# On the history of ring geometry
(with a thematical overview of literature)

Dirk Keppens

Faculty of Engineering Technology, KU Leuven
Gebr. Desmetstraat 1
B-9000 Ghent BELGIUM
<EMAIL_ADDRESS>
###### Abstract

In this survey paper we give an historical and at the same time thematical
overview of the development of “ring geometry” from its origin to the current
state of the art. A comprehensive up-to-date list of literature is added with
articles that treat ring geometry within the scope of incidence geometry.

In questo documento di ricerca forniamo una panoramica storica e allo stesso
tempo tematica dello sviluppo della “geometria sopra un anello” dalla sua
origine allo stato attuale. È aggiunto una lista di letteratura aggiornata
completa di articoli che trattano la geometria degli anelli nel contesto della
geometria dell’incidenza.

In diesem Forschungsartikel geben wir einen historischen und gleichzeitig
thematischen Überblick über die Entwicklung der “Ringgeometrie” von ihrem
Ursprung bis zum aktuellen Stand der Technik, mit einer Liste aktualisierter
Literatur einschließlich Artikeln zur Ringgeometrie im Kontext der
Inzidenzgeometrie.

Dans ce document de recherche, nous fournissons un aperçu historique et à la
fois thématique du développement de la “géométrie sur un anneau”, de son
origine à l’état actuel des connaissances. Nous ajoutons une liste de la
littérature actualisée comprenant des articles traitant la géométrie sur un
anneau dans le contexte de la géométrie de l’incidence.

Keywords: Ring geometry, projective ring plane, Hjelmslev geometry,
Klingenberg geometry, Barbilian plane, neighbor relation, bibliography

AMS Classification: Primary 51C05; Secondary 00A15, 01A60, 01A61, 51-03,
51A05, 51B05, 51E26, 51H10, 51F15

##  1 Introduction

The current version of the Mathematics Subject Classification (MSC), a
classification scheme used by the two major mathematical reviewing databases,
Mathematical Reviews (MathSciNet) and Zentralblatt MATH (zbMATH), provides
code 51C05 for all research papers dealing with Ring Geometry. This rather
small category contains articles about geometries that are not only provided
with an incidence relation but also with a neighbor relation (or its negation,
a non–neighbor or remoteness relation). Included are all geometries obtained
from rings that are not division rings. Ring geometry, i.e. the theory of
geometries equipped with a neighbor relation in general and of geometries over
rings in particular, is a rather young discipline. Its origin lies in the
beginning of the 20th century and its importance has grown steadily. For that
reason it has also got a full–fledged place as a chapter in the “Handbook of
Incidence Geometry” [A24]. In the past decades several mathematicians have
contributed to ring geometry. Newly discovered connections with coding theory,
with the theory of Tits–buildings and with quantum information theory, have
opened new horizons. In this survey paper we present an historical and
thematical overview with attention to many aspects. An out-of-date list with
articles on the subject, up to 1989, was composed by Törner and Veldkamp in
[A23] as an addition and completion of two even older lists [A02] and [A12]
written by respectively Artmann, Drake et al. and Jungnickel. Up to now such
list of literature is not available for the period after 1990, except for a
survey paper by the author [A13], dealing exclusively with plane projective
geometries over finite rings. In the present work we fill that gap and we add
a new updated list of existing literature, ordered thematically (and
containing the relevant material from the preceding lists). Articles on
algebraic geometry and on differential geometry over rings are not included.
We also do not aim for completeness when it concerns metric aspects, neither
for geometries on modular lattices nor for geometric algebra over rings. We
think that the bibliography might be useful for researchers who want to attack
the future challenges of ring geometry.

##  2 The first traces of ring geometry: dual numbers and Johannes Hjelmslev

The first traces of ring geometry date back to the beginning of the twentieth
century. The Danish geometer Johannes Trolle Hjelmslev (1873–1950) who was
born as Johannes Petersen but who changed his name in 1904, graduated in
mathematics from the University of Copenhagen and received his PhD degree in
1897. Hjelmslev can be viewed as one of the early founders of ring geometry.
In a series of four lectures, held at the University of Hamburg in July 1922
and published one year later in the Abhandlungen aus dem Mathematischen
Seminar [A10], he presented an axiomatic framework for geometry that better
reflected the properties observed in the real world. The basic observation
made by Hjelmslev was that, if one draws lines “close” to each other (meaning
that the sharp angle they define is very small), then it is hard to identify
the intersection point, and it looks as if the lines have a little segment in
common. Dually, if two points are close to each other, they belong to a line
segment that can be part of several joining lines. Hjelmslev called this Die
natürliche Geometrie, the “natural geometry”. In fact, Hjelmslev put forward
this idea already some years earlier, in 1916, in an article [A09] on what he
called Die Geometrie der Wirklichkeit, the “geometry of reality”.

In order to obtain a model for his geometry, Hjelmslev made use of the ring of
dual numbers of the form $a+b\varepsilon$ with $a$ and $b$ both real and
$\varepsilon^{2}=0$.
Dual numbers were already well–known before Hjelmslev. William Clifford
defined them for the first time in 1873 [A06] and they were used as a
convenient tool in mechanics by Aleksander Kotelnikov [A18] and by Eduard
Study in his famous work “Geometrie der Dynamen” [A22].
According to Benz [A05] the first traces of ring geometry can be observed
already in Study’s work and in that of some of his contemporaries like Josef
Grünwald, Pilo Predella and Corrado Segre. In their papers dual numbers are
treated from a geometrical viewpoint, by connecting the geometry of oriented
lines (spears) in the real euclidean space with a spherical geometry over the
ring of dual numbers [A08, A19, A21].

##  3 The pioneers of ring geometry: Barbilian and Klingenberg

Twenty years after Hjelmslev, Dan Barbilian (1895–1961), besides a
mathematician at the University of Bucharest, also one of the greatest
Romanian poets (with pseudonym Ion Barbu), took up the line again. In two
papers [A03, A04] (extended versions of a lecture hold in Baden–Baden in 1940)
he gave the first axiomatic foundation for plane projective ring geometry. He
started by investigating the conditions which must be imposed upon an
arbitrary associative ring in order that the corresponding geometry may have
“useful” properties. It turns out that the rings must have a unit element and
that all left singular elements must be two–sided singular or equivalently
$ab=1$ implies $ba=1$. Barbilian called these rings “Zweiseitig singuläre
Ringe” or $Z$–rings. Today such rings are also known as Dedekind–finite rings.
They include of course all commutative rings but also all finite rings (even
non–commutative) and many other classes of rings (e.g. matrix rings over a
field). Starting with $C$–rings (“Kategorische Ringe”), being $Z$–rings with
some additional property, Barbilian defined a kind of projective plane.
Conversely, he formulated a set of axioms for a plane geometry with incidence
and non–neighborship (“klare Lage”) and proved that this leads to a $C$–ring.
Barbilian’s work showed some shortcomings, but nevertheless it was of great
importance for the development of geometry over arbitrary rings as we will
discuss further (see also section 11).

Another monument among the ring geometers was the German mathematician Wilhelm
Klingenberg (1924–2010). He is best known for his work on differential
geometry, but in a series of papers [A14, A15, A16, A17], published in the
mid–fifties, he also laid the foundation for affine, metric (euclidean) and
projective geometries with neighbor relation (“mit Nachbarelementen”). His
central idea is the existence of a natural map from the geometry under
consideration onto an “underlying” ordinary geometry, a consequence of the
assumption that the neighbor relation is transitive. Therefore Klingenberg
called such geometries “Geometrien mit Homomorphismus”. He not only defined
them axiomatically, but he also constructed explicit examples using rings. The
assumption of a transitive neighbor relation is interlaced with the fact that
the rings must be local ones. A not necessarily commutative local ring $R$ is
a ring with a unique maximal left (or equivalently a unique maximal right)
ideal $R_{0}$. The quotient ring $R/R_{0}$ is a (skew)field, coordinatizing an
ordinary geometry which is the natural epimorphic image of the ring geometry
over $R$.

The work of Klingenberg was the source of two mainstreams in ring geometry in
the decades afterwards: Klingenberg geometry and Hjelmslev geometry.
Geometries with a transitive neighbor relation or equivalently with a
canonical map onto an ordinary geometry are now named Klingenberg geometries.
Their epimorphic image is obtained by considering the equivalence classes of
neighboring elements as “thick” elements. Non–neighboring elements behave like
distinct elements in the epimorphic image (e.g. two non–neighboring points are
incident with exactly one line in a projective Klingenberg plane since two
distinct points in its epimorphic projective plane are incident with a unique
line). No further assumptions are made about the neighboring elements.
Klingenberg geometries comprise the smaller, but more intensively studied
class of Hjelmslev geometries for which additional axioms must hold when it
comes to neighboring elements (e.g. two neighboring points can be connected by
at least two distinct lines in a projective Hjelmslev plane). This idea relies
on the natural geometry of Hjelmslev (see section 2). In the case of a
Klingenberg plane over a local ring $R$, the Hjelmslev condition means that
the ring must be a two–sided chain ring (a ring is a right chain ring if for
any $a$ and $b$ in $R$ either $a\in bR$ or $b\in aR$, similar for left chain
ring) with the additional property that its maximal ideal contains only zero
divisors. Such rings are called PH–rings (projective Hjelmslev rings or
$H$–rings). Finite chain rings are always $H$–rings and, equivalently, local
principal ideal rings. Klingenberg himself considered both Klingenberg planes
(over local rings) and Hjelmslev planes (over Hjelmslev rings) in [A16].

To be complete, we must mention also two isolated, rather unnoticed,
contributions to early geometry over rings. J.W. Archbold [A01] wrote on
projective geometry over group algebras $K[G]$ and worked out a small finite
example taking $K=$ GF(2) and $G$ a group of order 2, while Rimhak Ree in
[A20] considered projective spaces as the modular lattices $L(R,M)$ of all
sub–modules of a module $M$ over a ring $R$, in particular over a full matrix
ring.

##  4 The Belgian contribution to early ring geometry: projective geometry
over rings of matrices

The ideas of Barbilian concerning projective geometries over rings instead of
fields, got some followers in the sixties and early seventies. Among them we
find a lot of Belgian contributors who studied projective geometries over
rings of matrices. Since such rings are not local, the geometries are not
Klingenberg nor Hjelmslev geometries.

Julien Depunt studied in [B01] the projective line and in [B03, B04]
projective planes over the ring of ternions. This ring contains the elements
$a_{1}\varepsilon_{1}+a_{2}\varepsilon_{2}+a_{3}\varepsilon_{3}$ with
$a_{i}\in\mathbb{C}$ and $\varepsilon_{i}^{2}=\varepsilon_{i}$ ($i=1,2$),
$\varepsilon_{1}\varepsilon_{3}=\varepsilon_{3}=\varepsilon_{3}\varepsilon_{2}$
and all other products equal to $0$. The ring of ternions is isomorphic to the
ring of upper triangular matrices with elements in the complex field. The main
result of Depunt concerns the embedding of the plane over the ternions into
the 5–dimensional complex projective space.

Inspired by this work, Cléry Vanhelleputte could generalize the results for
planes over the full matrix ring of $2\times 2$–matrices with elements in an
arbitrary commutative field. His doctoral thesis was published in [B14] and
fitted in the research program under Julien Bilo (1914–2006) who was head of
the Geometry Department at the University of Ghent at that time.

A few years later, in 1969, Joseph Thas wrote his PhD thesis under the
supervision of Bilo about the projective line over the ring of $3\times
3$–matrices with elements in an algebraically closed field [B11], see also
[B10]. Soon afterwards Thas published [B12, B13] on projective ring geometry
over matrix rings, in particular over finite rings $M_{n}(GF(q))$ of $n\times
n$–matrices with entries in a Galois field of order $q$. Especially [B12] was
important as it contains some concepts (ovals, arcs, caps, etc.) and
combinatorial theorems which extend known results from classical Galois
geometry over finite fields and which appear for the first time for ring
geometries. Later Thas grew out to an authority in finite geometry, but his
interest shifted from finite ring geometry to Galois geometry and other
topics, in particular finite generalized quadrangles.
Thas was a colleague of Willem Mielants during many years. It was he who
guided Hendrik Van Maldeghem (see section 14) and the author towards ring
geometry as PhD supervisor for both of us.

Paul De Winne was the last student of Bilo studying ring geometry, see [B05].
His doctoral thesis was published in [B06].

Around the same time, Xavier Hubaut, another Belgian mathematician from the
Université Libre de Bruxelles, introduced the projective line over a general
associative algebra and investigated the structure of its group of
projectivities in [B08, B09]. His colleague Franz Bingen could generalize the
results to $n$–dimensional projective spaces over semi–primary rings in [B02].

Mind that the geometries over matrix rings studied by all these people, may
not be confused with the “geometries of matrices” initiated by the Chinese
mathematician Loo–Keng Hua in the mid–forties [B07]. Hua’s geometries of
matrices are not ring geometries in the narrow sense, despite the suggestive
name (although a connection is possible with projective lines over rings, see
section 13).

##  5 The foundations of plane affine ring geometry: from Benz to Leissner and
beyond

Barbilian, as well as all members of the Belgian School, dealt exclusively
with projective ring geometry. The first traces of affine geometries over
rings can be found in a paper from 1942 by Cornelius Everett, concerning
affine planes over rings without zero divisors [C16].
Walter Benz (1931–2017), a renowned German geometer, especially for his work
on circle geometries, considered in [C04, C05] affine planes over a special
kind of commutative rings and in [C06] plane affine (and metric–affine)
geometries over arbitrary rings with unit. A big part of this interesting
paper discusses the relation between algebraic properties of the ring and
geometric properties of the plane.

Peter Ashley Lawrence obtained a PhD under the supervision of Benz. His thesis
“Affine mappings in the geometries of algebras” considers ring geometries
associated with modules and algebras over a commutative ring and was published
in [C21]. Two other publications [C02, C03] by Hans–Joachim Arnold can be
classified within this section. They contain a simultaneous generalization of
affine geometry over rings and generalized affine spaces in the sense of
Emanuel Sperner [C41]. Arnold starts with the axiomatic definition of an
affine–line geometry which can be coordinatized by a vectorial groupoid and he
adds some extra axioms to turn it into a module over a unitary ring. Also
[C18] and [C34] both fit into this approach.

Affine planes over local rings, today also known as desarguesian affine
Klingenberg planes, appear as particular examples in the papers of Benz, but
this was not for the first time. Klingenberg already considered them on the
side of projective ring planes and in the more general context of “Affine
Ebenen mit Nachbarelementen” in the articles [A14, A15, A16, A17].
A reasonable part of the research concerning affine ring geometry focusses on
affine Hjelmslev planes (AH–planes). An important paper [C28] in that respect
was written by Heinz Lüneburg (1935–2009) in 1962. It contains the main
results of his doctoral thesis entitled “Affine Hjelmslev–Ebenen mit
transitiver Translationsgruppe”, dealing with translation AH–planes as
generalizations of ordinary translation planes. Werner Seier investigated in
more detail desarguesian and translation AH–planes in [C36, C37, C38, C39,
C40]. The central theme in his work is a characterization of desarguesian
AH–planes by an affine variant of Desargues’ theorem and the transitivity of
the set of translations in the plane. Some other papers on affine Hjelmslev
planes are not mentioned here and we postpone them to sections 7 and 8 where
they will be discussed in connection with projective Hjelmslev planes.

Without minimizing the importance of the papers cited above, we can state that
the real breakthrough of affine ring geometry in full generality was not
achieved yet at this point. Therefore one has to wait for Werner Leissner, a
student of Benz. Leissner wrote two papers in the mid–seventies on, what he
called, “affine Barbilian planes” (today the name “Leissner planes” would be
more convenient). In [C22] such planes are defined axiomatically as affine
structures in which two parallelodromy axioms hold (suitable substitutes for
the affine Desargues theorem) and they are coordinatized by an arbitrary
$Z$–ring $R$ together with a special subset $B$ of $R\times R$ (a Barbilian
domain). In [C23] the converse is proved: any affine ring plane over a
$Z$–ring is an affine Barbilian plane. Leissner considered even more general
affine Barbilian structures in [C24].
The papers of Leissner were a source of inspiration for further
generalizations by several people. Victoria Groze and Angela Vasiu considered
affine structures over arbitrary rings in [C17], Francisc Radó defined affine
Barbilian structures in [C33] by weakening Leissner’s axioms, Armentrout et
al. investigated in [C01] generalized affine planes which can be coordinatized
by near-rings and Pickert studied tactical configurations over quasirings in
[C32].

Other contributors to the theory of plane affine geometries with neighbor
relation and parallellism were Gernot Dorn [C13], Kvetoslav Burian [C08, C09,
C10], Frantisek Machala [C29, C30, C31], Stefan Schmidt and Ralph Steinitz
[C35], Franco Eugeni [C14, C15] and Angela Vasiu [C42, C43, C44]. The
Barbilian domains defined by Leissner also formed a study object in
themselves. Several authors have contributed to this, see [C07, C20, C25, C26,
C27]. For a recent and fairly complete overview of affine planes over finite
rings we refer to the survey paper [C19] of the author. We also mention here
two papers [C11, C12] by Basra Çelik on finite hyperbolic Klingenberg planes
as they are not far away from finite affine Klingenberg planes.

##  6 Metric geometry over rings and the school of Bachmann

The goal of this section is just to give a glimpse of metric ring geometry. We
do not aim for completeness when it concerns metric aspects, because they are
only indirectly related to incidence geometry. The literature about this
subject is extensive and we have selected only a few representative papers in
which much more references can be found. The basis of “modern” metric geometry
is provided by the work of several German mathematicians, including Bachmann,
Lingenberg and Schröder.
The pivotal figure is Friedrich Bachmann (1909–1982). In the second edition of
his famous book “Aufbau der Geometrie aus dem Spiegelungsbegriff” [D01] he
considers plane metric geometries in which points, lines, incidence and
orthogonality are defined by means of a group of reflections. A group $G$ with
a subset $S$ of involutory elements which are invariant under inner
automorphisms of $G$ and which generate $G$, determines a group plane
$E=E(G,S)$ as follows: the elements of $S$ are the lines of $E$ and the
involutory elements of $S^{2}$ are the points. Two lines that commute are
perpendicular. A point and a line that commute are incident. If the group
plane $E$ satisfies the conditions (1) a point and a line determine a unique
perpendicular, and (2) the product of three lines that are either concurrent
or have a common perpendicular is again a line, then the pair $(G,S)$ is
called a Hjelmslev group and the associated group plane $E$ is called a metric
(non–elliptic) Hjelmslev plane.
If the uniqueness of the line incident with two distinct points is not
required, then metric Hjelmslev planes stripped of their metric structure,
reduce to incidence Hjelmslev planes. In section 2 we have already discussed
the role of Hjelmslev as founder of (incidence) ring geometry. In some later
work, the “Algemeine Kongruenzlehre” [D06], Hjelmslev used special
transformations (reflections) to define orthogonality. Hence, the rudiments of
metric ring geometry also go back to Hjelmslev and have been worked out in
more detail by Bachmann. There is also a minor contribution by Klingenberg who
treats metric aspects in ring geometries in [A15].

Rolf Lingenberg (1929–1978), a student of Bachmann, continues his work in
[D12]. The main result is an algebraic characterization of classical metric
group planes $E(G,S)$ with an additional axiom, as planes associated with a
metric vectorspace $(V,q)$ with $q$ a quadratic form.
Eberhard Schröder in [D18, D19] considers metric planes of different kind
starting from a pappian affine plane over a field and using twodimensional
algebras for the introduction of the metric notions of angle, distance and
orthogonality. His work is strongly related with circle geometries studied
intensively by Benz (see section 13). A good overview of classical metric
geometry (but not including ring geometries) can be found in Chapter 17
written by Schröder in the Handbook of Incidence Geometry [D20].

There are numerous contributions by Bachmann and his school to the theory of
Hjelmslev groups, connecting geometric and algebraic properties of groups and
planes. From Bachmann himself we mention the important additions [D02, D03,
D04] to his book [D01]. In the seventies and eighties his successors, most of
them from the University of Kiel (Germany), have extended the knowledge.
Finite Hjelmslev groups were characterized by Rolf Stölting in his PhD thesis
“Endliche Hjelmslev-Gruppen und Erweiterungen von Hjelmslev-Gruppen”, also
published in [D21]. In [D22] Hjelmslev groups are constructed from a module
$M$ over a commutative ring $R$ endowed with a bilinear form.
Edzard Salow introduced singular Hjelmslev groups (in which the product of
three points is always a point) in his doctoral thesis “Beiträge zur Theorie
der Hjelmslev-Gruppen: Homomorphismen und Singuläre Hjelmslev-Gruppen”
published in [D15]. The main result is the construction of a coordinate ring
$R$ for the group plane of a singular Hjelmslev group, proving that these
metric planes are indeed ring geometries. The process of algebraization of
metric Hjelmslev planes is investigated also in [D16]. It is proved there that
a Hjelmslev group with some additional axioms can be embedded in the
orthogonal group of a metric module $(R^{3},f)$ with $R$ a commutative ring
with unit for which $2$ and any non zero–divisor is invertible and with $f$ a
symmetric bilinear form of the free module $R^{3}$.
In [D17] Salow studies another class of metric ring planes using a commutative
algebra over a ring and the concept of an angle. In an early paper of Benz
[C04] the metric notion of angle also played an important role and in [C06] he
devotes a paragraph to metric geometry, using an elliptic form over an
arbitrary commutative ring.
Similar work can be found in [D13, D14] in which Wolfgang Nolte proves that a
class of metric planes $E(G,S)$ with additional axioms can be embedded into a
projective Hjelmslev plane over a local ring and that $G$ is isomorphic to a
subgroup of an orthogonal group. Other results in this direction were obtained
by Frieder Knüppel, in [D07, D08, D09, D10, D27], Gerald Fischbach [D05],
Michael Kunze [D11] and Rolf and Horst Struve [D23, D24, D25, D26].
The influence of Bachmann is also clear from the large amount of doctoral
theses produced in Germany on Hjelmslev groups and metric geometry, e.g. R.
Schnabel (1974), M. Kunze (1975), H. Struve (1979), R. Struve (1979), M.
Gegenwart (1987), W. Vonck (1988) and A. Bach (1998).

##  7 The florescence of Hjelmslev geometry in the era of Drake, Artmann and
Törner

The first period of florescence of ring geometry (especially Hjelmslev
geometry) regarded as incidence geometry, started in the late sixties and
reached his culmination point in the seventies. This is reflected in a large
number of publications. Two mathematicians who were very productive in this
area and left their mark, were David Allyn Drake (1937–) from the University
of Florida, Gainesville (USA) and Benno Artmann (1933–2010) from the
University of Giessen (Germany).

Drake obtained his PhD in 1967 with a thesis entitled “Neighborhood
collineations and affine projective extensions of Hjelmslev planes” under the
supervision of Erwin Kleinfeld. Kleinfeld himself, an authority in algebra,
with a lot of publications on non–associative alternative rings, published
only one, though interesting paper [E25] about ring geometry. It was the first
publication concerning finite Hjelmslev planes. Among other things he
introduced a two parameter set $(s,t)$ of non–zero integers such that for each
flag $(P,\ell)$ in a finite projective Hjelmslev plane, there are exactly $t$
points on $\ell$ neighboring with $P$ and exactly $s$ points on $\ell$ not
neighboring with $P$. It was proved that $s\leq t^{2}$ or $t=1$. If $s=t^{2}$,
the plane is called uniform. In that case all point neighborhoods have the
structure of ordinary affine planes. Robert Craig proved in [E11] that any
finite projective plane can be extended to a uniform projective Hjelmslev
plane.
The notion of uniformity (and its generalization to $n$–uniformity) has played
a crucial role in the work of Drake. It was also related to another issue: the
extension of an affine Hjelmslev plane to a projective Hjelmslev plane. An
ordinary affine plane can always be extended to a projective plane, but for
Hjelmslev planes the situation is much more complicated. Drake has written
several papers about this problem in the period from 1968 to 1975. In [E14] he
proves that any uniform affine Hjelmslev plane has at least one (uniform)
projective extension and he gives an example of a desarguesian uniform affine
Hjelmslev plane with a non–desarguesian projective extension. In [E15]
uniformity is generalized to $n$–uniformity inductively (a PH–plane is
$n$–uniform if the point neighborhoods are $(n-1)$-uniform AH–planes).
Strongly $n$–uniform planes are characterized by a local property, which leads
to the theorem: an $n$–uniform PH–plane is strongly $n$–uniform if and only if
its dual is $n$–uniform. Drake also proved that every finite desarguesian
PH–plane is strongly $n$–uniform.
A further study of $n$–uniform Hjelmslev planes (projective and affine) was
made in [E16, E17, E18, E19, E20, E21, E22] where even more general affine
geometries with neighbor relation appear. Drake could also prove that there do
exist affine Hjelmslev planes which cannot be extended to projective Hjelmslev
planes.

Artmann was a contemporary of Drake and he wrote his doctoral thesis
“Automorphismen und Koordinaten bei ebenen Verbänden” in 1965 under the
supervision of Günther Pickert. In his early work on Hjelmslev geometry we can
observe a strong relation with the theory of modular lattices. In [E01] he
gives a sufficient condition for a modular lattice to define a projective
Hjelmslev plane and in [E03] he proves that any uniform PH–plane can be
derived from a modular lattice.
Like Drake, Artmann also studies refinements of the neighbor relation
(“verfeinerten Nachbarschaftsrelationen”) and the affine–projective extension
question. In [E04] he proves that a uniform affine Hjelmslev plane can be
extended to at least two non–isomorphic projective Hjelmslev planes. A new
concept introduced by him in [E02] is that of a projective Hjelmslev plane of
level $n$ (“$n$–ter Stufe”) based on the refinement of the neighbor relation.
This definition was extended by Drake to the affine case in [E20]. Artmann
proves that desarguesian PH–planes over a Hjelmslev ring $R$ are of level $n$
if and only if the maximal ideal of $R$ is nilpotent of index $n$, see [E05,
E06].

Another theorem proved by Artmann [E07] states that for any projective plane
$\mathcal{P}$ and any integer $n>0$ there exists a PH–plane of level $n$ with
$\mathcal{P}$ as epimorphic image. Moreover, given a sequence of PH–planes
$\ldots\rightarrow H_{i}\rightarrow H_{i-1}\rightarrow\ldots\rightarrow
H_{1}=\mathcal{P}$ with $H_{i}$ of level $i$, the inverse limit is a
projective plane. Arno Cronheim constructed in [E12] in a purely algebraic
way, using formal power series over a cartesian group, a chain of Hjelmslev
planes whose inverse limit is a projective plane. Cronheim also obtained a
complete classification of all finite uniform desarguesian projective
Hjelmslev planes in [E13]. They are either planes over a ring of twisted dual
numbers over GF($q$) (a non–commutative generalization of the classical dual
numbers) or over a truncated Witt ring $W_{2}(q)$ of length 2.

Both Drake and Artmann had a great influence on the mathematical research in
the domain of Hjelmslev geometry (even when Artmann’s interest shifted to
other subjects after a few years). One of Drake’s students was Phyrne Bacon.
She wrote both her Master’s thesis “On Hjelmslev planes with small invariants”
and her PhD thesis “Coordinatized Hjelmslev planes” on Hjelmslev geometry,
resulting in two papers [E08] and [E09]. She proved that a finite Hjelmslev
plane is strongly $n$–uniform if and only if it is of level $n$ which unifies
the two notions introduced by Drake and Artmann respectively. Later, Bacon’s
attention shifted to the more general Klingenberg geometries (see section 9).
Artmann was the supervisor of Manfred Dugas and Günther Törner who both made
important contributions to Hjelmslev geometry. The PhD thesis of Dugas
“Charakterisierungen endlicher desarguescher uniformer Hjelmslev-Ebenen”
contains many new ideas, including a coordinatization method (see section 9).
In [E23] Dugas proves that a finite translation AH–plane can be extended to a
PH–plane if it is or a desarguesian PH–plane or an ordinary translation
projective plane, while in [E24] he gives a necessary and sufficient condition
for a projective Hjelmslev plane to be derivable from a lattice. In particular
the PH–planes of level $n$ are always lattice–derivable.

Törner wrote his Master’s thesis on “Hjelmslev–Ringe und die Geometrie der
Nachbarschaftsbereiche in der zugehörigen Ebenen” and obtained his PhD with
“Eine Klassifizierung von Hjelmslev-Ringen und Hjelmslev-Ebenen” in the same
year as Dugas, under the supervision of Artmann and Pickert. His research in
the domain of ring geometry focusses on two main themes: the structure of
(finite) Hjelmslev planes and the ideal structure of chain rings. Among his
publications we mention here [E36] which contains the main results from his
thesis: a classification of PH–planes based on congruence relations. He proves
that the set of all congruence relations of a finite PH–plane is linearly
ordered under inclusion and consequently, the canonical epimorphism onto the
associated projective plane admits an essentially unique factorization into
indecomposable epimorphisms. The plane $\mathcal{H}$ is of “type $n$” or
“height $n$” if the canonical epimorphism $\varphi$ from $\mathcal{H}$ onto
the projective plane $\overline{\mathcal{H}}$ has a maximal factorization
$\mathcal{H}=\mathcal{H}_{n}\rightarrow\mathcal{H}_{n-1}\rightarrow\ldots\rightarrow\mathcal{H}_{1}=\overline{\mathcal{H}}$.
In [E40, E41] Törner investigates the equivalence of finite $n$–uniform planes
or planes of level $n$ as defined by Drake and Artmann to planes of type $n$.
In [E39] he proves that $n$–uniform projective Hjelmslev planes are strongly
$n$–uniform. Some of the results were later extended to the infinite case in
[E43]. In [E40] much attention goes also to affine Hjelmslev planes, in
particular translation AH–planes over near–rings. In Törner’s work
homomorphisms play an important role as can also be seen from [E37, E38].

In the desarguesian case (Hjelmslev planes over a chain ring) the structure of
the plane is intrinsically connected with the ideal structure of the ring. The
structure of chain rings and valuation rings was investigated by Törner partly
in collaboration with Hans–Heinrich Brungs. One of their papers [E10] concerns
the embedding of right chain rings into chain rings (related with the problem
of embedding desarguesian affine Hjelmslev planes into projective ones), see
also [E32, E33, E35, E42].
With a postdoctoral scholarship Artmann stayed for a short time at the
McMaster University in Ontario (Canada). There he inspired J.W. (Mike) Lorimer
who would later become one of the leading figures in topological Hjelmslev
geometry (see section 10). In [E31] Lorimer and Lane study desarguesian
Hjelmslev planes. They prove that an affine Hjelmslev plane is desarguesian if
and only if it can be coordinatized by an AH–ring and that not every
desarguesian AH–plane can be extended to a desarguesian PH–plane. Morphisms
between affine Hjelmslev planes are the main subject in [E26, E27, E30] while
[E28, E29] deal with the structure of Hjelmslev rings.

##  8 The continuation of the Hjelmslev epoch under Drake, Jungnickel and Sane

In his publications on $n$–uniform planes, we can observe already that Drake
had a particular interest in finite Hjelmslev planes. This is continued when
his attention goes more and more to the problem of existence and non–existence
of finite Hjelmslev planes with given parameters. In a series of papers [F05,
F06, F07, F08, F13, F16, F20], some of them with co–author, he attacked this
problem and he linked finite PH–planes to nets in [F10, F19].
Meanwhile, also finite Klingenberg planes came to the attention [F09]. Drake
and Lenz considered a parameter set for finite PK–planes in [F15] together
with new examples of finite PH–planes.
Structure theorems for finite chain rings (needed for finite desarguesian
Klingenberg planes) were proved by Edwin Clark et al. in [F03, F04] and
independently by Arnold Neumaier in [F33] and Al–Khamees [F01]. The
classification of all chain rings is still an open problem but partial results
are known. Galois rings GR($q^{n},p^{n}$) with $q^{n}$ elements and
characteristic $p^{n}$, with $q=p^{r}$, play a crucial role.

Beside Drake another player came to the forefront, Dieter Jungnickel, who was
active at the Universities of Giessen and Augsburg (Germany). He was a student
of Hanfried Lenz and became an expert in the theory of designs. In 1976 he
wrote his Diplomarbeit at the University of Berlin (Germany) on “Klingenberg
and Hjelmslev planes” and with the dissertation “Konstruktion transitiver
Inzidenzstrukturen mit Differenzenverfahren” he obtained his doctoral degree.
His most important contribution to the theory of Hjelmslev planes (and the
more general class of Klingenberg planes) concerns “regularity” [F14, F23,
F24, F26, F29]. A PK– or PH–plane is regular if it has an abelian automorphism
group $G=Z\oplus N$, where $G$ acts regularly (sharply transitively) on the
point set and on the line set and $N$ acts regularly on each neighborhood. It
is proved in [F21] that any finite PK–plane over a commutative local ring is
regular. Regularity is also interpreted in terms of difference sets and
auxiliary matrices, leading to new families of finite Hjelmslev and
Klingenberg planes. An interesting result, connecting PH–planes with designs,
is: the projective uniform Hjelmslev planes of order $q$ (with $q>2$) are
precisely the symmetric divisible partial designs on two classes with
parameters
$v=b=q^{2}(q^{2}+q+1),k=r=q(q+1),s=q^{2},t=q^{2}+q+1,\lambda_{1}=q,\lambda_{2}=1$.
For $q=2$ counterexamples exist, see [F27]. The concepts of regularity and
uniformity were also considered in $K$–structures, a further generalization of
Klingenberg planes (see [F09, F11, F12, F22, F25, F28, F30, F31]). Nino
Civolani [F02] considers free extensions of partial Klingenberg planes.

Jungnickel’s work was continued by Sharad Sane. Sane studied at the Indian
Institute of Technology Bombay, Mumbai (India) and obtained his PhD with the
dissertation “Studies in Partial Designs and Projective Hjelmslev Planes”
under the supervision of Balmohan Vishnu Limaye. In [F32] Sane and Limaye
demonstrate that $n$–uniform PH–planes are a kind of divisible partial designs
and by taking advantage of this property, they can give an alternative proof
for the fact that $n$–uniform planes are strongly $n$–uniform, as was proved
before in another way by Törner [E39]. In some other papers [F17, F18, F34,
F35, F36, F37, F38] Sane contributes to the theory of finite Hjelmslev and
Klingenberg planes.

##  9 The coordinatization of Hjelmslev and Klingenberg planes: a versatile
story

The coordinatization of affine and projective planes is one of the most
powerful tools in the study of such geometries. It permits to reformulate
geometric properties into algebraic ones (and vice versa), leading to a better
insight, including the construction of many non–desarguesian examples. This
coordinatization goes back to Marshall Hall Jr. (1910–1990). His important
paper [G37] published in 1943 is still one of the most cited. The basic
concept is a Hall ternary ring, also called PTR (planar ternary ring), an
algebraic structure $(R,T)$ with $R$ a non–empty set containing two distinct
elements $0$ and $1$ and with $T$ a ternary operation on $R$ such that
$y=T(x,m,k)$ means that the point with coordinates $(x,y)$ lies on the line
with coordinates $[m,k]$. With $(R,T)$ one can associate two loops $(R,+)$ and
$(R,\circ)$ when defining $a+b:=T(a,1,b)$ and $a\circ b:=T(a,b,0)$. The
properties of the plane (formulated in terms of the validity of Desargues’
configuration or in terms of transitivity of the automorphism group) are
reflected in the richness of the coordinatizing algebraic structure. If the
theorem of Desargues is always valid or equivalently if the plane is
$(P,\ell)$–transitive for any choice of the point $P$ and the line $\ell$,
then it turns out that $(R,+)$ and $(R\setminus\\{0\\},\circ)$ are both
groups, hence $(R,+,\circ)$ is a division ring or skewfield. Conversely, any
skewfield gives rise to a desarguesian projective plane. Slight variations on
Hall’s coordinatization method were made by Daniel Hughes [G38, G39] and by
Günter Pickert [G55]. Independently the russian mathematician Lev Anatolevich
Skornyakov described in 1949 a similar coordinatization method in [G58]. His
work [G59] was important for the distribution of the knowledge on projective
planes in the russian speaking mathematical community.

In the seventies and the eighties several attempts were made to coordinatize
in a similar way affine and projective Hjelmslev and Klingenberg planes. In
1967 the russian geometer V.K. Cyganova worked out a first successful
coordinatization for the more restrictive class of affine Hjelmslev planes
[G21]. She used the concept of an $H$–ternar, an algebraic structure with two
ternary operations, generalizing a Hall ternary ring (one of the main
differences being the existence of zero divisors). Since her paper was written
in russian, it remained unfortunately unaccessible for many people.
Independently from Cyganova, J.W. Lorimer introduced in 1971 in his PhD thesis
“Hjelmslev Planes and Topological Hjelmslev Planes” generalized ternary rings
(very similar to $H$–ternars) as the coordinatizing structures of affine
Hjelmslev planes. Three years later, Phyrne Bacon streamlined the work of
Cyganova and Lorimer in her thesis “Coordinatized Hjelmslev planes”, and she
introduced the name biternary ring (in appendix A of her thesis she gives a
comprehensive list of annotations including some mistakes and imperfections in
the work of her predecessors). The interaction between the geometric
properties of an affine Hjelmslev plane and the algebraic properties of its
coordinatizing biternary ring was examined in more detail by Lorimer in [G46],
by Cyganova in [G20, G22, G23, G24, G25, G26, G27], by Emelchenkov in [G33,
G35, G36] and by Shatokhin in [G56, G57].
To be complete, we also have to mention a paper by Drake [G28] in which he
obtains a kind of coordinatization for a special class of affine Hjelmslev
planes (radial planes) by means of a module.

After the coordinatization of affine Hjelmslev planes, a similar theory for
the more general class of affine Klingenberg planes has been worked out by
several authors. Bacon generalized her biternary rings. Drake, her supervisor,
encouraged her to publish in mathematical journals, but she was stubborn and,
apart from one single publication [G05], she refused. Her voluminous work,
totaling about 1000 pages, is contained in four books [G04], edited in own
management. For that reason it was often overlooked and seldom recognized as
an acceptable reference. In [G05] the “triangle theorem” is proved: a PK–plane
$\mathcal{P}$ possessing a nondegenerate triangle with sides
$\ell_{1},\ell_{2}$ and $\ell_{3}$ such that each derived AK–plane
$\mathcal{A}_{i}=\mathcal{P}\setminus g_{i}$ is desarguesian, is itself a
desarguesian PK–plane.
The Czech mathematician Frantisek Machala from the University of Olomouc
introduced in [G49] affine local ternary rings as an alternative for the
coordinatization of affine Klingenberg planes. Much later he could prove the
equivalence between his coordinatization method and the one of Bacon. He also
proved that any “incomplete” biternary ring (with one ternary and one partial
ternary operator) can be extended to a biternary ring with two (full) ternary
operators [G52]. In [G11] it is shown that this biternary ring extension is
unique.

In the case of ordinary planes each (desarguesian) affine plane can be
extended to a (desarguesian) projective plane. This does not hold any longer
for Hjelmslev planes (see section 7). This observation has also a serious
impact on the coordinatization of projective Hjelmslev planes: it does not
follow immediately from the affine coordinatization. The projective case was
first attacked by the russian mathematician E.P. Emelchenkov in 1972 in his
PhD thesis ”Ternars and automorphisms of Hjelmslev planes” (in russian) and in
[G34]. Due to the language barrier, his work was not accessible for many
researchers and for that reason, like Cyganova’s work, it was somewhat
overlooked.
Coordinatization methods for the more general case of projective Klingenberg
planes were worked out by both Machala and Bacon. Their approach is totally
different. Machala’s method is based on the concept of an extended local
ternary ring, an algebraic structure $(R,R^{\prime},T)$ with two disjoint sets
of coordinates $R$ and $R^{\prime}$ and one ternary operation $T$ on $R\cup
R^{\prime}$ (see [G47, G48, G50]). This coordinatization was not very
successful because it is not obvious to see any interaction between properties
of the extended local ternary ring and geometric properties of the
coordinatized plane.
The approach of Bacon is based on the fact that a PK–plane can be covered by
three AK–planes corresponding to three biternary rings. This yields a
coordinatizing structure for a projective Klingenberg plane as a triplet of
biternary rings, called sexternary ring [G04]. In Bacon’s voluminous work, the
interaction between geometric properties and the algebraic structure is
examined in depth. Unfortunately, a big part of these results remained hidden
for the reason mentioned above. In [G31] Manfred Dugas used a similar
coordinatizing structure, with six ternary operations.
Independently from the people mentioned above, the author introduced in 1987
in his PhD thesis “Klingenberg incidence structures, a contribution to ring
geometry” (in Dutch) (see also [G41, G42, G43]) planar sexternary rings
(PSR’s) with one full and five partial ternary operators. His coordinatization
method for PK–planes was inspired by the Hughes variant of the Hall ternary
ring. A small deficiency in his method was detected later (as pointed out by
Baker and Lorimer in [G11]). As a consequence of this shortcoming, the
coordinatization of a PK–plane by a PSR wasn’t fully compatible with the
coordinatization of a derived AK–plane by the biternary ring obtained from the
PSR. To overcome this anomaly, Baker and Lorimer (op.cit.) developed a new
coordinate ring, called incomplete sexternary ring (in the spirit of Dugas) as
a substitute for the planar sexternary ring. They even proved that such a
structure can be extended (in a unique way) to a sexternary ring with six full
ternary operators.

The coordinatization of projective planes is a handy instrument for the
construction of non–desarguesian examples in an algebraic manner. A lot of new
planes were found using quasifields, nearfields or alternative division rings.
Because of the bigger complexity of sexternary rings it seems that much less
examples of non–desarguesian PK–planes were obtained in this manner.
Nevertheless examples of non–desarguesian AK– and PK–planes obtained from
algebraic structures, are known. The oldest examples are the Moulton affine
Hjelmslev planes, given by Baker in [G06]. A projective version is contructed
in [G43] by the author. Klingenberg planes over local nearrings and Hjelmslev
planes over Hjelmslev–nearrings were studied by Emanuel Kolb in [G44, G45].
Much attention has gone to Moufang planes which can be coordinatized by local
alternative rings. Moufang–Hjelmslev planes first appear in a paper of Dugas
[G29]. He proves that all finite uniform Moufang PH–planes are desarguesian. A
stronger version of that theorem was later proved in [G30] (the uniformity
condition could be dropped if the order of the plane is bigger than 2).
Similar results were found by Baker, Lane and Lorimer for Moufang Klingenberg
planes, see [G07, G09, G10]. They prove that the class of Moufang PK–planes
coincides with the class of planes over local alternative rings and that a
finite Moufang PK–plane in which any two points have at least one joining
line, is a desarguesian projective Hjelmslev plane. Also a stronger version of
Bacon’s triangle theorem was proved in [G08]: a PK–plane with a non–degenerate
triangle for which the three derived AK–planes are translation AK–planes (and
with epimorphic image distinct from PG(2,2)), is Moufang.
More recently, a group of mathematicians around Basri Çelik and Süleyman
Çiftçi, from the University of Uludag, Bursa (Turkey), published several
papers concerning a particular class of Moufang–Klingenberg planes [G01, G02,
G14, G03, G15, G16, G17, G18, G19]. Their results, all variations on the same
theme, overlap with work of Andrea Blunck [G12, G13]. The role of Pappus’
theorem (its validity in a desarguesian plane implies the commutativity of the
coordinatizing ring) was investigated by Nolte and Maurer in [G53, G54].

##  10 Order and topology in Hjelmslev geometry: Machala and Lorimer

The theory of ordered incidence structures can be traced back mainly to Pasch,
Vorlesungen über neuere Geometrie from 1882. An excellent survey paper on the
axiomatics of ordered incidence geometry is [H27].
Ordered affine and projective Hjelmslev planes were studied by a group of
Canadian mathematicians, starting in the seventies. At least three
dissertations were written in that period at the McMaster University of
Hamilton, Ontario (Canada) under supervision of Norman Lane who was also a
world–class canoeist, competing in two Olympic games (bronze medal in 1948 in
London), before he started his academic career.
In 1975, Lynda Ann Thomas, wrote her Master’s thesis on “Ordered desarguesian
affine Hjelmslev planes” in which she proved that any ordered AH–ring gives
rise to an ordered desarguesian affine Hjelmslev plane and vice versa. This
result was published a few years later in [H30].
James Laxton, another student of Lane, treated in his Master’s thesis “Ordered
non–desarguesian affine Hjelmslev planes”. Catherine Baker, also a student of
Lane, wrote her Master’s thesis on “Affine Hjelmslev and generalized affine
Hjelmslev planes” and her doctoral thesis in 1978 on “Ordered Hjelmslev
planes”. In that thesis she investigates in detail the relationship between
ordered AH–planes and the coordinatizing ordered biternary rings, extending
results of Laxton and Thomas. Baker published several papers about ordered
(affine and projective) Hjelmslev planes (also with co–author): [H01, H02,
H03, H04, H05, H06].
It would be disrespectful if we should attribute all the results on ordered
ring geometries to the “Canadian School” only. Independently, a theory of
orderings for Klingenberg planes was worked out by Machala. He published many
papers on ordered Klingenberg planes [H18, H19, H20, H21, H22, H23, H24, H25]
and one overview work [H26]. The work of Baker et al. resembles in many
aspects Machala’s, but there are some differences. For a comparison between
both approaches, one may consult [H06].

The study of topological Klingenberg and Hjelmslev planes remained an
exclusive Canadian affair. The most prominent student of Lane, was undoubtedly
J.W. (Mike) Lorimer. In his doctoral thesis on “Hjelmslev planes and
topological Hjelmslev planes”, he not only introduced a coordinatization (see
the previous section) but he also laid the foundation for topological
Hjelmslev geometry. His work generalizes that of Salzmann [H28] and Skornyakov
[H29] on topological projective planes. In a series of publications [H07, H08,
H09, H10, H11, H12, H13, H14, H15, H16, H17] he further developed the theory
in close connection with the coordinatization problem. Among the most
important theorems proved by Lorimer, we mention the following
characterization theorem: the only locally compact connected pappian
projective Hjelmslev planes are the ones over the rings $\mathbb{K}[x]/\langle
x^{n}\rangle$ with $\mathbb{K}$ the field of real or complex numbers.

##  11 The revival of ring geometry in the eighties and nineties under
Veldkamp and Faulkner

In the seventies ring geometry was restricted almost exclusively to Hjelmslev
and Klingenberg geometry (in the desarguesian case to geometries over local
rings and Hjelmslev rings). The Dutch mathematician Ferdinand Douwe Veldkamp
(1931-1999), who is well–known for his work on geometries associated with
exceptional Lie groups and in particular polar spaces, reverted back to the
pioneering work of Barbilian where geometries over the broader class of
$Z$–rings were considered. It was Veldkamp’s aim to give an axiom system for
projective planes (and higherdimensional spaces) over arbitrary rings with
unit (without the imperfection in Barbilian’s attempt [A03, A04]). From
conversations with his colleague van der Kallen at the University of Utrecht
(an expert in $K$–theory), it became clear that the best setting for this
project is provided by rings of stable rank two. A ring $R$ has stable rank
two if the following property holds: if $a,b\in R$ and $Ra+Rb=R$ then there
exists a $r$ in $R$ such that $a+rb$ is invertible in $R$. The class of stable
rank two rings comprises all local rings. Hence, the projective ring planes
introduced by Veldkamp include the desarguesian Klingenberg and Hjelmslev
planes. A ring of stable rank two is always a $Z$–ring.
Veldkamp first worked out the theory for planes in [I23] with some special
cases in [I24] and later for spaces of higher dimension (see section 12). The
ring planes defined by Veldkamp are also known today as desarguesian Veldkamp
planes.

John Robert Faulkner, an authority in the domain of non–associative algebra
and geometry, further extended the theory of Veldkamp planes in the
non–desarguesian direction by introducing alternative (non–associative) rings
of stable rank two in [I01]. He then proved in [I02] that a Veldkamp plane has
the Moufang property (i.e. $(P,\ell)$–transitivity holds for all $P,\ell$ with
$P$ incident with $\ell$) if and only if it is a plane
$\mathcal{P}(\mathbb{A})$ over an alternative ring $\mathbb{A}$ of stable rank
two. Inspired by the work of his predecessors, Faulkner introduced in [I03]
Faulkner planes as a very general class of plane incidence structures with
neighbor (or remoteness) relation. They comprise the Veldkamp planes and the
planes introduced by Barbilian. A (connected) Faulkner plane for which the
group of $(P,\ell)$–transvections (automorphisms fixing all objects incident
with $P$ or $\ell$) is transitive on the set of points not neighboring with
$\ell$, is called a transvection plane. The coordinatization of transvection
Faulkner planes by a not necessarily associative alternative ring with the
property that $ab=1$ implies $ba=1$ involves a rather technical procedure
based on group theory and a lot of new concepts such as covering planes and
tangent bundle planes. A transvection Faulkner plane for which the tangent
bundle plane is also a transvection plane is called a Lie transvection
Faulkner plane. To every such plane an alternative two–sided units ring can be
attached and conversely with an alternative two–sided units ring $R$ a
corresponding Lie transvection Faulkner plane can be constructed. However,
this plane is not determined unambiguously when the ring does not have stable
rank 2. This high price has to be paid for the generalization from Veldkamp
planes to Faulkner planes. In [I04] Faulkner gives a geometric construction of
Barbilian planes coordinatized by composition algebras (including the Moufang
plane) using Jordan algebras. His book [I07] is completely devoted to the role
of such algebras in projective geometry.
Faulkner was surrounded by students at the University of Virginia,
Charlottesville (USA), who all were involved with the study of ring
geometries. Terese Deltz Magnus obtained her PhD in 1991 on “Geometries over
non–division rings” and she could generalize Faulkner’s axioms and results for
geometries of higher dimension (see section 12). Eve Torrence also graduated
under Faulkner’s supervision with “The coordinatization of a
hexagonal–Barbilian plane by a quadratic Jordan algebra”, a generalization of
the classical notion of generalized hexagon. Another student was Catherine
Moore d’Ortona who studied homomorphisms between projective ring planes in her
PhD thesis “Homomorphisms of remotely projective planes”, published in [I16].
Finally Karen Klintworth wrote her PhD thesis on “Affine remoteness planes”.

Faulkner himself considered a slightly more general axiomatization of Faulkner
planes in [I06]. There he chooses for the remoteness relation, the negation of
the neighbor relation. Much of the results obtained in [I03] are extended but
the coordinate rings that appear are no longer always alternative. It is
proved that $\mathcal{P}$ is a transvection plane if and only if $\mathcal{P}$
is isomorphic to $P(G,N)$, the plane associated with a group $G$ of Steinberg
type parametrized by the ring $R$ and with $N$ a certain subgroup of $G$.
Necessary and sufficient conditions are given for $R$ to be alternative,
associative or commutative. In [I06] also projective remoteness planes with
reflections (hence metric planes) have been considered. The content of this
paper is closely related to some work of Knüppel and Salow in [D10] (see
section 6). It also contains a part on affine ring planes and elementary basis
sets which are closely related to Barbilian domains as introduced by Leissner
(see section 5).

In the slipstream of Veldkamp’s paper on projective ring planes, several
slightly modified axiom systems have been described, leading to other classes
of projective ring geometries. We have already mentioned Frieder Knüppel in
the section on metric ring geometry, but some of his papers rather join the
spirit of this section. In [I14] Knüppel considers ring geometries over
associative rings based on four axioms adapted from Veldkamp (based on
remoteness rather than neighborship). A coordinatization theorem is stated
without proof. In [I15] he studies homomorphisms between such planes.
Renata Spanicciati defines near–Barbilian planes and strong near–Barbilian
planes in [I22] by adapting some of the axioms of Veldkamp. The neighbour
relation between points turns out to be the identity, and the neighbour
relation between lines becomes an equivalence relation. In [I11] Guy Hanssens
and Hendrik Van Maldeghem prove that any near–Barbilian plane is strong
near–Barbilian. Kálman Péntek gives a necessary and sufficient condition for a
Veldkamp plane to be a direct product of a finite number of Veldkamp planes in
[I17, I18, I19].
The study of homomorphisms between ring geometries was also a central theme in
several papers of Veldkamp (some of them in joint work with Joseph Ferrar)
[I08, I09, I10, I25, I26]. The geometric homomorphisms of distinct kind
(incidence preserving, neighbor–preserving, distant–preserving) are
characterized in terms of algebraic morphisms between the underlying rings.
Veldkamp’s results were generalized by the author in the non–desarguesian case
for homomorphisms between projective Klingenberg planes using the
coordinatizing planar sexternary rings (see section 9) in [I12, I13].
Thorsten Pfeiffer could generalize a well–known theorem for planes over fields
to planes over rings: a desarguesian ring plane $\mathcal{P}(R)$ is pappian
(Pappus’ theorem is valid) if and only if $R$ is commutative.

##  12 Projective and affine Hjelmslev spaces and spaces over arbitrary rings

Hitherto we only discussed plane ring geometries. The theory of higher
dimensional projective spaces over rings has been developed by different
people. Projective spaces over local rings appear for the first time in the
work of Klingenberg [A17]. Today we call them Klingenberg spaces (PK–spaces).
The first study of PK–spaces after Klingenberg, is due to Hans–Heinrich Lück,
a student of Lüneburg. His PhD thesis, published as an article [J34] in 1970
under the somewhat misleading title “Projektive Hjelmslevräume”, contains an
axiomatic characterization of a class of incidence structures which permit a
coordinatization by local rings. Hence, the paper deals with projective
Klingenberg spaces rather than with Hjelmslev spaces. Lück proves that in an
axiomatically defined PK–space of dimension at least three, the theorem of
Desargues holds and that it must be isomorphic to a space derived from a
module over a local ring. Hence all projective Klingenberg spaces of dimension
$\geq 3$ are desarguesian, a situation analogous to the case of classical
projective spaces.
Independently from Lück, Machala defined and studied projective Klingenberg
spaces (Projektive Räume mit Homomorphismus) of finite or infinite dimension
in [J35, J36, J37, J38, J39]. He proved that the planes in a PK–space are
PK–planes and that PK–spaces of dimension at least three come from modules
over a local ring (cfr. Lück). Machala also investigated homomorphisms between
PK–spaces and the fundamental theorem (isomorphisms between spaces can be
represented by semilinear mappings between the underlying modules). A paper by
Jukl [J13] is in conformity with this.
An axiomatic approach for the more restricted class of projective Hjelmslev
spaces (PH–spaces) was initiated by John Lamb Jr. from the University of Texas
at Austin (USA) in his PhD thesis, entitled “The Structure of Hjelmslev space,
a generalization of projective space” (1969) but its content (related to
lattice theory) was not published.
Much more widespread is the work of Karl Mathiak, who was very productive in
this field. He defined a class of special projective Hjelmslev spaces starting
from a vectorspace over a (skew)field endowed with a valuation (“Bewertete
Vektorräume”). The structure of the ideals in the corresponding valuation ring
plays a central role in his approach. The theory is thoroughly worked out in a
series of papers, published over a period of twenty years between 1967 and
1987, see [J41, J42, J43, J44, J45, J46, J47, J48, J49, J50].
His compatriot Alexander Kreuzer introduced an axiom system for arbitrary
PH–spaces in his doctoral thesis “Projektive Hjelmslevräume” which was
published afterwards as an article in [J18] with some preliminary work in
[J17]. This study is continued in [J19, J20].

For the affine case we have to go to Canada again. In four papers, Tibor
Bisztriczky together with J.W. Lorimer, worked out two axiom systems for
affine Klingenberg spaces [J02, J03, J04, J05]. Neither of their axiom systems
assumes the existence of an overlying projective Klingenberg space or the
existence of an underlying ordinary affine space. Machala in [J38] also
defined affine Klingenberg spaces, but not separate from PK–spaces (similar to
ordinary affine spaces obtained from projective spaces by deleting a
hyperplane).

The most general study of projective ring spaces (over not necessarily local
rings) was undertaken by Ferdinand Veldkamp. We have already indicated his
interest in section 11. In [J62, J63], Veldkamp gives a self–dual axiom system
for projective “Barbilian spaces” of finite dimension using the basic concepts
of points, hyperplanes, incidence and neighbor relation. We call them now
Veldkamp spaces. The main result is that Veldkamp spaces of dimension $\geq 3$
are spaces over rings of stable rank 2.
Theresa Magnus defines Faulkner spaces as even more general geometries by
extending the theory of Faulkner planes [J40]. She proves that any Faulkner
space of dimension $n\geq 3$ is coordinatized by a unique associative
two–sided units ring $R$ and that the group generated by all transvections is
a group of Steinberg type over $R$. A Faulkner space over the ring
$\mathbb{Z}$ of integers is constructed, providing an example of a Faulkner
geometry which is not a Veldkamp space, since $\mathbb{Z}$ has stable rank 3.
Under the Veldkamp spaces we also find the projective spaces over matrix rings
over GF($q$), studied by Thas [B12] in the early days of ring geometry (see
section 4). Other contributions to finite ring spaces are due to Kapralova who
considered projective spaces over the ring of dual numbers over a Galois field
in [J14] and to Ivan Landjev and Peter Vandendriesche [J24, J25].

A well-developed theory for affine spaces over rings is also due to Veldkamp.
In [J64] he defines Barbilian domains in free modules of rank $n$ and
introduces $n$–dimensional affine ring geometries. A geometrical
interpretation of Barbilian domains is given by Sprenger in [J61].
Other attempts for setting up a theory of higher dimensional affine ring
geometries (incidence structures with parallellism) are scattered in the
literature. The definitions and the methods used are very diverse.
Contributions in this field are due Permutti and Pizzarello [J54, J55], Miron
[J51], Leissner, Severin and Wolf [J31, J32], Ostrowski [J53], Schmidt and
Weller [J58], Kreis [J15, J16], Seier [J59, J60], Bach [J01] and others.

One of the problems for higher dimensional geometries over rings that has got
much attention is that of morphisms and the fundamental theorem. For classical
projective geometries $P(V)$ induced by a vectorspace over a field this
theorem states that any bijective incidence preserving map (projectivity)
between projective spaces $P(V)$ and $P(W)$ can be algebraically characterized
by a semilinear map from $V$ to $W$. The first generalization of the
fundamental theorem to ring geometries was obtained by the Indian
mathematicians Ojanguren and Sridharan who could prove it in case of
module–induced geometries $P(M)$ with $M$ a free module of finite rank $\geq
3$ over a commutative ring [J52]. Generalizations to other classes of rings
were proved later by Sarath and Varadarajan in [J57] and by James in [J12] and
Faure in [J06].
For the sake of completeness we also refer to module–induced geometries as
defined by Marcus Greferath and Stefan Schmidt [J08, J09, J10, J11]. Instead
of the usual definition of the pointset as the set of all submodules generated
by a unimodular element in a free module (cfr. Veldkamp), they take all
submodules of rank one, leading to the bizarre situation of points properly
contained in bigger points. In [J07] an extension of the fundamental theorem
is proved for such module–induced geometries. Close relative to this, there is
an abundance of articles by the school of the Georgian mathematician Alexander
Lashkhi. They all contain variations on the same theme: an extension of the
fundamental theorem for affine and projective geometries related with modules
over rings, from the lattice–theoretic point of view. We have not included the
whole collection of papers by Lashki and his students. Some of them have been
published multiple times in different journals (in Russian and in English).
The literature list only mentions a few representative ones: [J21, J22, J23,
J26, J27, J28, J29, J30, J56].

##  13 Projective lines and circle geometries over rings: Blunck, Havlicek and
Keppens

The “smallest” projective geometries that can be considered, are the
projective lines. From the viewpoint of incidence geometry not much can be
said about these rather poor structures. But combining the study of projective
lines with those of their automorphisms (the general linear group) puts them
into a new light. The theory has common ground with what is usually called
geometric algebra. Indeed, the projective line $P(R)$ over any ring $R$ can be
defined in terms of the free left $R$–module $R^{2}$ as follows: it is the
orbit of a starter point $R(1,0)$ under the action of the general linear group
$GL_{2}(R)$ on $R^{2}$. Since geometric algebra over rings only fits sideways
in the section of incidence geometry, we did not make an effort to be complete
in the literature list for this item. Nevertheless we mention a number of
relevant references in which more information can be found, e.g. [K46].
Central themes that keep returning are the notions of cross–ratio and harmonic
quadruples and the fundamental theorem, also known as Von Staudt’s theorem. In
the case of classical projective lines over a field or a skewfield, this
theorem characterizes mappings of the projective line which preserve
harmonicity as projectivities.

Among the first publications on projective ring lines (and we do not consider
here papers only dealing with linear groups over rings) belong some articles
by the Indian mathematicians Nirmala and Balmohan Limaye. They prove a
generalization of Von Staudt’s theorem for some special classes of commutative
and non–commutative rings in [K38, K39, K40, K41, K42]. A little bit earlier,
in 1968, Melchior, a student of Benz, wrote his PhD thesis, entitled “Die
projektive Gerade über einem lokalen Ring: ihre lineare Gruppe und ihre
Geometrie”. Other contributions to this item appeared in [K02, K04, K05, K10,
K16, K22, K23, K24, K25, K30, K31, K34, K35, K37, K45, K47, K49]. Some papers
by Bilo and Depunt [B01], Hubaut [B08, B09] and Thas [B10, B11] also deal with
projective lines over rings (see section 4). They were followed by Havlicek et
al. in [K26, K27, K28].

Projective lines over rings are also intrinsically related with circle
geometries. This relation was established for the first time by Benz in his
famous book “Vorlesungen über Geometrie der Algebren” [K06] from 1973. He
presented a unified treatment of plane circle geometries, now called Benz
planes, using the projective line over a commutative ring which is a
two–dimensional $\mathbb{K}$–algebra over a field $\mathbb{K}$. His definition
was extended by Andrea Blunck, a student of Benz, and Armin Herzer who
introduced more general chain geometries $\Sigma(\mathbb{K},R)$ with $R$ a
(not necessarily two–dimensional) $\mathbb{K}$–algebra. A chain geometry is an
incidence structure whose point set is the set of points of the projective
line over $R$ and the $GL_{2}(R)$ orbit of $P(\mathbb{K})$ is the set of
chains. The plane circle geometries of Möbius, Laguerre and Minkowski type are
particular chain geometries for $R=\mathbb{L}$ (a quadratic field extension of
$\mathbb{K}$), $R=\mathbb{K}+\mathbb{K}\varepsilon$ with $\varepsilon^{2}=0$
(dual numbers) or $R=\mathbb{K}+\mathbb{K}t$ with $t^{2}=t$ (double numbers)
respectively. For an overview of chain geometry we refer to [K19]. Chain
geometries were also treated by Schaeffer, a student of Benz, in his doctoral
thesis entitled “Zum Automorphismenproblem in affinen Geometrien und
Kettengeometrien über Ringen. The study of chain geometries is continued by
Blunck who introduced generalized chain geometries by considering projective
lines over non–associative, alternative rings. In her PhD thesis
“Doppelverhältnisse und lokale Alternativringe” (1990) and in [K07] she
extended the notion of cross–ratio and investigated chain geometries (in
relation to projective lines over non–associative rings) in [K08, K11, K12].

Around the same time the author in [K32, K33] defined Klingenberg–Benz planes
axiomatically, i.e. plane circle geometries with neighbor relation which admit
a natural epimorphism onto a classical Benz plane. Using the projective line
over three kinds of quadratic ring extensions of a local ring (instead of a
field), he was able to construct algebraic models of such geometries. Also
Konrad Lang in [K36] studied independently of us a class of Hjelmslev–Möbius
planes. Blunck and Stroppel extended our definition of Klingenberg–Benz planes
to Klingenberg chain spaces in [K21] and Blunck also proved that a Klingenberg
chain space can be embedded into a projective Klingenberg space, such that the
points are identified with points of a quadric and the chains with plane
sections [K09]. In [K50] Seier constructed analogously a class of chain
geometries $\Sigma(H,L)$ with $H$ a Hjelmslev–ring and $L$ a ring extension of
$H$ and in [K51] he defined a Möbius plane with neighbor relation of a
different kind than the one defined by us.
A basic notion concerning the projective line over a ring $R$ is its distant
relation: two points are called distant if they can be represented by the
elements of a two-element basis of $R^{2}$. The distant graph has as vertices
the points of the projective line and as edges the pairs of distant points.
The distant graph is connected precisely when $GL_{2}(R)$ is generated by the
elementary linear group $E_{2}(R)$. This aspect of projective ring lines was
studied in more detail by Blunck, Havlicek, Matraś and some others in [K01,
K13, K14, K15, K29, K43, K44]. In [K17, K18] the interaction between ring
geometry and the geometry of matrices in the sense of Hua (see [B07]) is
investigated in more detail.

##  14 Ring geometries and buildings: Van Maldeghem and co.

The theory of buildings was invented by the Belgium-born French mathematician
Jacques Tits. Roughly speaking, buildings are incidence geometries on which
groups act. Tits received many awards for his fundamental and path–breaking
mathematical ideas, including the Abel Prize in 2008. One of his achievements
is the classification of affine buildings of rank at least 4. They are known
to be “classical”, i.e. they arise from algebraic groups over a local field.
In the rank three case (where affine buildings are of three possible types
$\tilde{A}_{2}$, $\tilde{C}_{2}$ or $\tilde{G}_{2}$) many non–classical
counterexamples are known.
In his PhD thesis “Niet–klassieke driehoeksgebouwen” on triangle buildings
(affine buildings of type $\tilde{A}_{2}$) Hendrik Van Maldeghem observed that
a special kind of ring geometry is present as the suitably defined geometry at
distance $n$ from any given vertex of the building (the so-called $n$–th
floor). This was described among other things in [L18, L19]. A little bit
later Hanssens and Van Maldeghem could prove that those ring geometries are in
fact projective Hjelmslev planes of level $n$, see [L07]. In [L08] they give a
universal construction for level $n$ Hjelmslev planes (see also [L06] for the
$2$–uniform case) and as a corollary any level $n$ projective Hjelmslev plane
is isomorphic to the $n$–th floor of a triangle building. This result links
the theory of PH–planes to that of triangle buildings, a rather unexpected but
fascinating fact. In the same spirit Van Maldeghem investigated another class
of rank three affine buildings, of type $\tilde{C}_{2}$, and proved that the
$n$–th floor turns out to be another type of ring geometry which can be seen
as a generalization of an ordinary generalized quadrangle. He called it
“Hjelmslev–quadrangle” of level $n$ (see [L20, L21]). In joint work with
Hanssens a complete characterization of $\tilde{C}_{2}$–buildings by Hjelmslev
quadrangles was obtained [L09, L10].
The author defined “Klingenberg–quadrangles” as another generalization of
ordinary generalized quadrangles in [L12]. The connection between
Klingenberg–quadrangles and Hjelmslev–quadrangles is explained in [L20].

The relation of polar spaces of higher rank to generalized quadrangles is
comparable with the relation of projective spaces to projective planes.
Generalized quadrangles are polar spaces of rank two. Projective ring spaces
of dimension at least three have got as much attention in the literature as
projective ring planes. This is not the case so far for general
“Klingenberg–polar spaces” or “polar spaces over rings” if the rank is bigger
than two. Only one paper by James [L11] is known to us. Certainly this topic
offers perspectives for future research.

The discovery by Van Maldeghem of the connection between buildings and ring
geometry has a precedent. Twenty years earlier, in 1968, Veldkamp in joint
work with Tonny Springer considered a geometry over the split octonions (over
the complex number field) in [L16]. This geometry is a kind of analogue of the
non–desarguesian projective plane over the alternative division ring of
(non–split) octonions $\mathbb{O}$ but in which two distinct lines may have
more than one point in common and dually. It was called Moufang–Hjelmslev
plane, but this name is misleading since it is not a projective Hjelmslev
plane (the neighbor relation is not transitive) and hence it is completely
distinct from the Moufang PH–plane (over an alternative local ring) studied
elsewhere (see section 9). In two subsequent papers [L25, L26] more results on
Hjelmslev–Moufang planes are obtained, concerning projective groups. The
geometry of Veldkamp and Springer is the same as the one constructed by Tits
[L17] starting from the split algebraic group of type $E_{6}$. In this
geometry each line has the structure of a polar space and two lines can meet
in more than one point (namely, in a maximal singular subspace of the
corresponding polar spaces).
John Faulkner considered Hjelmslev–Moufang planes by allowing an arbitrary
ground field instead of $\mathbb{C}$ in [L04, L05] and also Robert Bix studied
generalized Moufang planes in [L02, L03].
The relationship between Hjelmslev planes and buildings was further exploited
by Van Maldeghem and Van Steen to give a characterization of some rank three
buildings by automorphism groups [L22, L23, L24].

In the margins of the study of buildings some other questions have emerged.
One of them concerns embeddings. Embeddings of point–line geometries into
projective spaces are well–known in the literature. The embedding question for
ring geometries, in particular for projective Hjelmslev planes, is first
attacked by Artmann [L01] who shows that the PH–plane over the ring of plural
numbers $\mathbb{F}[t]/t^{n}$ ($\mathbb{F}$ a field), can be embedded in the
$(3n-1)$–dimensional projective space over $\mathbb{F}$. In [L13] the author
and Van Maldeghem prove a nice characterization theorem for embeddable
Klingenberg planes: if $\mathcal{P}$ is a projective Klingenberg plane that is
fully embedded in the projective space PG(5,$\mathbb{K}$) for some skewfield
$\mathbb{K}$, then $\mathcal{P}$ is either a desarguesian Klingenberg plane
over a ring of twisted dual numbers or a subgeometry of an ordinary projective
plane. The embedding of the projective plane over a matrix ring with entries
in GF($q$) into a projective space over GF($q$) was also observed by Thas in
[B12, B13].
Veronesean sets are closely connected with embeddings. In [L14] Schillewaert
and Van Maldeghem define geometries with an additional axiom by which the
Hjelmslev–Moufang plane (in the sense of Springer–Veldkamp) and its relatives
fit into the framework using the modern notion of parapolar spaces. In [L15]
they provide a common characterisation of projective planes over two-
dimensional quadratic algebras (over an arbitrary field) in terms of
associated Veronesean sets. Anneleen De Schepper and Van Maldeghem [L27] have
considered Veronese representations of Hjelmslev planes over quadratic
alternative algebras as part of a more general study of Veronese varieties and
Mazzocca–Melone sets.

##  15 Ring geometries in coding theory: Honold, Kiermaier and Landjev

One of the fastest growing disciplines in mathematics is coding theory. Since
its introduction by Claude Shannon in 1948, the number of publications about
codes has exploded, in particular due to its importance in cryptography, data
transmission and data storage. Initially mostly linear codes over finite
fields were studied, but after the publication in 1994 of the paper [M07] by
Hammons _et al._ , a new era has begun. In that paper it is proved that all
(non–linear) binary Kerdock-, Preparata-, Goethals- and Delsarte-Goethals-
codes are images of $\mathbb{Z}_{4}$–linear codes under the Gray map. This
discovery was quite peculiar and the paper got in 1995 the Information Theory
Paper Award from the IEEE Information Theory Society. It was the start of a
search for new codes by considering linear codes over the ring
$\mathbb{Z}_{4}$ and over more general finite rings (see e.g. [M01]). Also
some papers by Aleksandr Nechaev [M47, M48] have led the research in that
direction.
We will not give a survey of all results obtained up to now for codes over
finite rings, because even this niche has become too wide. We restrict
ourselves here (and in the literature list) to the publications in which the
direct relation between codes over rings and ring geometries is exhibited.
Indeed, linear codes over finite chain rings can be associated with finite
projective Hjelmslev geometries, the hyperplanes corresponding to the
codewords. This correspondence offers opportunities for investigating the
structure and the construction of ring–linear codes by pure geometrical
methods.
The technique was first applied in [M14] by Thomas Honold, now working at the
Zhejiang Universität in Hangzhou (China), and Ivan Landjev from the New
Bulgarian University of Sofia (Bulgaria). They prove that certain MacDonald
codes can be represented by linear codes over the ring of twisted dual numbers
on a finite field, using multisets of points in Hjelmslev spaces. In [M15]
they prove that all Reed–Muller codes are linearly representable over the ring
of dual numbers over $\mathbb{Z}_{2}$. In [M16] a general theory of linear
codes over finite chain rings has been developed as a natural generalization
of the theory of linear codes over finite fields and the correspondence with
Hjelmslev spaces is investigated. In [M17] and [M18] an update of that paper
is given. Geometric arguments are also used explicitly in [M19] for the
construction of particular linear codes over chain rings of order four,
generalizing a result obtained by Michael Kiermaier and Johannes Zwanzger in
[M34, M35]. Keisuke Shiromoto and Leo Storme defined in [M49] a Griesmer type
bound for linear codes over finite quasi-Frobenius rings and they give a
geometrical characterization of linear codes meeting the bound, viz. a one-to-
one correspondence between these codes and minihypers in projective Hjelmslev
spaces.
Kiermaier was a student of Honold and wrote his Master’s thesis on “Arcs und
Codes über endlichen Kettenringen” in 2006 and obtained his PhD in 2012 at the
University of Bayreuth (Germany) with the thesis “Geometrische Konstruktionen
linearer Codes über Galois-Ringen der Charakteristik 4 von hoher homogener
Minimaldistanz”.

The study of codes over chain rings from the viewpoint of Hjelmslev geometry
has also led to the generalization of several special point sets (arcs, ovals,
blocking sets, caps), already well–known in classical geometry over fields.
The thus far most studied sets in Hjelmslev planes are arcs. A $(k,n)$-arc
refers to a set of $k$ points which meet every line in at most $n$ points.
This definition was given by Honold and Kiermaier in [M43]. General upper
bounds on the cardinality of such arcs were found as well as the maximum
possible size. For a chain ring $R$ of length 2 with Jacobson radical $R_{0}$,
such that $|R/R_{0}|=q$, the maximum size of a $(k,2)$-arc in the projective
Hjelmslev plane over $R$ is $q^{2}$ if $q$ is odd and $q^{2}+q+1$ if $q$ is
even (see [M20]). In [M21] the existence of maximal $(q^{2}+q+1,2)$-arcs (i.e.
hyperovals) is proved for $q$ even and in [M12] the existence of maximal
$(q^{2},2)$-arcs for $q$ odd is proved. The results on maximal arcs are also
used to construct interesting codes with a linear representation over a chain
ring. Examples, non–existence results and upper bounds for the length of arcs
are also present in [M02, M08, M10, M11, M13, M23, M32, M33, M36, M50].
Caps in finite projective Hjelmslev spaces over chain rings of nilpotency
index 2 are defined by Honold and Landjev in [M22]. A geometric construction
for caps in the threedimensional space is given, using ovoids in the
epimorphic space PG(3,$q$) as well as an algebraic construction using
Teichmüller sets. Blocking sets in Hjelmslev planes and their relation with
codes is the subject of [M03, M40, M42] while [M39] and [M54] deal with
spreads.

Another aspect that appears in the literature is the correspondence between
two–weight codes and strongly regular graphs. In [M04] regular projective two-
weight codes over finite Frobenius rings are introduced and it is shown that
such codes give rise to a strongly regular graph. In [M04, M41] two-weight
codes are constructed using ring geometries and they yield infinite families
of strongly regular graphs with non-trivial parameters.

Ovals in an ordinary projective plane of order $q$ are just $q+1$ arcs and
every conic is an oval. By a celebrated theorem of Segre every $(q+1)$–arc in
PG(2,$q$) with $q$ odd, is a conic. The study of ovals, conics and unitals in
finite projective Klingenberg and Hjelmslev planes was initiated by the author
in his PhD thesis but only a part of it was published, e.g. in [M31]. The
author also defined polarities in projective Klingenberg planes and spaces and
he investigated their sets of absolute points which give rise to ovals or
ovoids in some cases. A comprehensive study of polarities in $n$–uniform
Hjelmslev planes and spaces over the ring GF($q$)[$t$]/$t^{n}$ appeared in
[M28, M29, M30]. Also the papers [M37, M38] enlight some aspects of ovals and
conics in ring planes.
A combinatorial study of conics in finite desarguesian Hjelmslev planes was
made by Ratislav Jurga and Viliam Chvál. Their papers contain formulas for the
number of interior and exterior points, tangents and secants of a conic [M05,
M25, M26, M27]. A related problem is the projective equivalence of quadrics in
projective Klingenberg spaces. This was first formulated in [M06] and analyzed
in detail by O.A. Starikova et al. in [M24, M44, M51, M52, M53].
Recently, a special class of codes (toric codes) is found to be related to
affine ring geometries (Leissner planes) by Little in [M45, M46], revealing
still another correspondence between ring geometry and coding theory.

##  16 Ring geometry in quantum information theory: Saniga and Planat

Besides the fact that ring geometries play a substantial role in coding
theory, there is another domain in which they arise unexpectedly, namely in
quantum information theory. In this important branch of quantum physics it is
studied how information can be stored and retrieved from a quantum mechanical
system. In 2006 Metod Saniga from the Astronomical Institute at the Slovak
Academy of Sciences and Michel Planat from the Université de Franche–Comté
(France) discovered a connection between finite ring geometry and quantum
information theory [N09]. It is not clear yet whether or not the
correspondence goes further than just equality in numbers of objects, but
anyhow it is remarkable that ring geometries seem to play a role in quantum
physics.

The notion of mutually unbiased bases (MUBs) has turned into a cornerstone of
the modern theory. Saniga and Planat observed that the basic combinatorial
properties of a complete set of MUBs of a $q$–dimensional Hilbert space
$\mathcal{H}_{q}$ with $q=p^{r}$, $p$ being a prime and $r$ a positive
integer, are qualitatively mimicked by the configuration of points lying on a
proper conic in a projective Hjelmslev plane defined over a Galois ring of
characteristic $p^{2}$ and rank $r$. The $q$ vectors of a basis of
$\mathcal{H}_{q}$ correspond to the $q$ points in a neighbour class and the
$q+1$ MUBs answer to the total number of pairwise disjoint neighbour classes
on the conic. In a series of subsequently published papers, other
combinatorial correspondences between concepts from quantum theory and ring
geometries over finite rings (in particular projective ring lines) are
observed. One of these similarities concerns the structure of the generalized
Pauli group associated with a specific single $d$-dimensional qudit (qudits
are generalizations tot $d$–level quantum systems of qubits which are the
basic units in quantum information systems of level two). See [N01, N02, N03,
N04, N06, N08, N11, N12, N13, N14, N15, N16, N17, N18] and the survey papers
[N05, N10] for an overview on this topic. The study of the relation with
finite geometry (not only restricted to ring geometry but also in connection
with small generalized polygons and polar spaces) is still ongoing, see e.g.
[N07].

##  17 Literature on ring geometry and geometry over rings

In the separate bibliographic list we only mention publications in scientific
journals (or books). Other references, e.g. Master’s and PhD theses which are
quoted explicitly in the foregoing text, are not repeated here, except in case
they were also published. Articles belonging to conference proceedings are
also omitted if there exist copies with almost identical content, published in
other journals. We have grouped the references in themes, corresponding to the
sections in the paper. Some papers can be classified in multiple sections. In
that case they are listed in the section in which they are referred to for the
first time. We do not claim that the bibliography is complete but we have
tried to compose an accurate list which complements the existing obsolete
lists, especially for the period after 1990. The author will be grateful for
additions, completions and corrections to this list.

## A First traces and pioneers of ring geometry

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  * [A03] Barbilian D., Zur Axiomatik der projektiven ebenen Ringgeometrien I, Jahresber. Deutsch. Math.-Verein. 50, 179–229 (1940)
  * [A04] Barbilian D., Zur Axiomatik der projektiven ebenen Ringgeometrien II, Jahresber. Deutsch. Math.-Verein. 51, 34–76 (1941)
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  * [A07] Dembowski P. Finite Geometries, reprint of the 1968 Edition, Springer-Verlag, Berlin Heidelberg, 379 pp. (1997)
  * [A08] Grünwald J., Über duale Zahlen und ihre Anwendung in der Geometrie, Monatsh. Math. 17, 81–136 (1906)
  * [A09] Hjelmslev J., Die Geometrie der Wirklichkeit, Acta Math. 40, 35–66 (1916)
  * [A10] Hjelmslev J., Die natürliche Geometrie, Abh. Math. Sem. Univ. Hamburg 2, 1–36 (1923)
  * [A11] Iordănescu R., The geometrical Barbilian’s work from a modern point of view, Balkan J. Geom. Appl. 1, 31–36 (1996)
  * [A12] Jungnickel D., Hjelmslev’sche Inzidenzgeometrie und verwandte Gebiete. Literaturverzeichnis II, J. Geom. 16, 138–147 (1981)
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  * [A14] Klingenberg W., Projektive und affine Ebenen mit Nachbarelementen, Math. Z. 60, 384–406 (1954)
  * [A15] Klingenberg W., Euklidische Ebenen mit Nachbarelementen, Math. Z. 61, 1–25 (1954)
  * [A16] Klingenberg W., Desarguessche Ebenen mit Nachbarelementen, Abh. Math. Sem. Univ. Hamburg 20, 97–111 (1955)
  * [A17] Klingenberg W., Projektive Geometrien mit Homomorphismus, Math. Ann. 132, 180–200 (1956)
  * [A18] Kotelnikov A. P., Screw Calculus and Some Applications to Geometry and Mechanics (in Russian), Annals of the Imperial University of Kazan, Russia (1895)
  * [A19] Predella P., Saggio di Geometria non-Archimedea (Nota II), Batt. G. 3, 161–171 (1912)
  * [A20] Ree R., On projective geometry over full matrix rings, Trans. Amer. Math. Soc. 6, 144–150 (1955)
  * [A21] Segre C., Le geometrie proiettive nei campi di numeri duali, Nota I e II, Atti R. Acc. Scienze Torino 47, 114–133 and 164–185 (1911–1912) (in: Corrado Segre, Opere, a cura della Unione Matematica Italiana, Volume II, Edizione Cremonese, Roma, 396–431 (1958))
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  * [A24] Veldkamp F.D., Geometry over rings, chapter 19 in Handbook of Incidence Geometry: Buildings and Foundations. Edited by F. Buekenhout, Elsevier, Amsterdam, 1420 pp. (1995)

## B The Belgian contribution to early ring geometry

  * [B01] Bilo J. and Depunt J., Over de lineaire transformaties van de ternionenrechte (in Dutch), Med. Koninkl. Vlaamse Acad. Wet., Lett. schone Kunsten België, Kl. Wet. 24 (8), 1–36 (1962)
  * [B02] Bingen F., Géométrie projective sur un anneau semi–primaire, Acad. Roy. Belg. Bull. Cl. Sci. 52, 13–24 (1966)
  * [B03] Depunt J., Sur la géométrie ternionienne dans le plan, Bull. Soc. Math. Belg. 11, 123–133 (1959)
  * [B04] Depunt J., Grondslagen van de analytische projectieve ternionenmeetkunde van het platte vlak (in Dutch), Verh. Kon. Vl. Ac. voor Wet., Lett. en Sch. K. van België, Kl. der Wet. 63, 1–99 (1960)
  * [B05] De Winne P., An extension of the notion “cross-ratio of an ordered 4-tuple of points of the projective line” to an ordered ($n+3$)-tuple of points (resp. hyperplanes) of the $n$-dimensional space over an arbitrary ring with identity, part I: the $n$-dimensional projective space $S_{n}$ over an arbitrary ring with identity, Simon Stevin, 47, 139–159 (1974)
  * [B06] De Winne P., Een studie betreffende het projectieve vlak over de totale matrixalgebra $M_{2}((K)$ der $2\times 2$–matrices met elementen in een algebraïsch afgesloten veld (in Dutch), Verh. Kon. Vl. Ac. voor Wet., Lett. en Sch. K. van België, Kl. der Wet., 144, 1–80 (1978)
  * [B07] Hua L.-K., Geometries of Matrices I. Generalizations of Von Staudt’s theorem, Trans. Amer. Math. Soc. 57, 441–481 (1945)
  * [B08] Hubaut X., Construction d’une droite projective sur une algèbre associative, Acad. Roy. Belg. Bull. Cl. Sci., 50, 618–623 (1964).
  * [B09] Hubaut X., Algèbres projectives, Bull. Soc. Math. Belg. 17, 495–502 (1965)
  * [B10] Thas J.A., Dubbelverhouding van een geordend puntenviertal op de projectieve rechte over een associatieve algebra met éénelement (in Dutch), Simon Stevin, 42 (3), 97–111 (1969)
  * [B11] Thas J.A., Een studie betreffende de projectieve rechte over de totale matrix algebra $M_{3}(K)$ der $3\times 3$–matrices met elementen in een algebraïsch afgesloten veld $K$ (in Dutch), Verh. Kon. Vl. Ac. voor Wet., Lett. en Sch. K. van België, Kl. der Wet., 112, 1–151 (1969)
  * [B12] Thas J.A., The $m$-dimensional projective space $S_{m}(M_{n}(GF(q)))$ over the total matrix algebra $M_{n}(GF(q))$ of the $n\times n$-matrices with elements in the Galois field $GF(q)$, Rend. Mat. 4, 459–532 (1971)
  * [B13] Thas J.A., Deduction of properties, valid in the projective space $S_{3n-1}(K)$, using the projective plane over the total matrix algebra $M_{n}(K)$, Simon Stevin, 46, 3–16 (1972)
  * [B14] Vanhelleputte C., Een studie betreffende de projectieve meetkunde over de ring der $(2\times 2)$-matrices met elementen in een commutatief lichaam, Verhdl. Vlaamse Acad. Wet., Lett. schone Kunsten België, Kl. Wet. (in Dutch) 92, 1–93 (1966)

## C The foundations of plane affine ring geometry

  * [C01] Armentrout N., Hardy F. and Maxson C., On generalized affine planes, J. Geom. 4, 143–159 (1974)
  * [C02] Arnold H-J., Die Geometrie der Ringe im Rahmen allgemeiner affiner Strukturen, Hamburger Mathematische Einzelschriften, Göttingen 4, 86 pp. (1971)
  * [C03] Arnold H-J., A way to the geometry of rings, J. Geom. 1, 155–167 (1971)
  * [C04] Benz W., Süssche Gruppen in affinen Ebenen mit Nachbarelementen und allgemeineren Strukturen, Abh. Math. Sem. Univ. Hamburg 26, 83–101 (1963)
  * [C05] Benz W., $\Omega$–Geometrie und Geometrie von Hjelmslev, Math. Ann. 164, 118–123 (1966)
  * [C06] Benz W., Ebene Geometrie über einem Ring, Math. Nachr. 59, 163–193 (1974)
  * [C07] Benz W., On Barbilian domains over commutative rings, J. Geom. 12, 146–151 (1979)
  * [C08] Burian K., Affine H-structures, Comment. Math. Univ. Carolin. 13, 629–635 (1972)
  * [C09] Burian K., Affine parallel $H$–structures (in Czech), Sb. Prací Ped. Fak. v Ostrav$\breve{\rm e}$ Ser. A 9, 3–5 (1974)
  * [C10] Burian K., Translation $H$–structures, Sb. Prací Ped. Fak. v Ostrav$\breve{\rm e}$ Ser. A Mat. Fyz. 21, 15–25 (1986)
  * [C11] Çelik B., On the hyperbolic Klingenberg plane classes constructed by deleting subplanes, J. Inequal. Appl. 357, 1–6 (2013)
  * [C12] Çelik B., A hyperbolic characterization of projective Klingenberg planes, Int. J. Math. Sci. 2, 10–14 (2008)
  * [C13] Dorn G., Affine Geometrie über Matrizenringen, Mitt. Math. Sem. Giessen 109, 120 pp. (1974)
  * [C14] Eugeni F. and Galiè E., Sui piani costruiti su anelli, Dipartimento M.E.T., Università di Teramo, Italy, 143–162 (1991)
  * [C15] Eugeni F. and Maturo A., Generalized affine planes, J. Inform. Optim. Sci. 12, 431–439 (1991)
  * [C16] Everett C.J., Affine geometries of vector spaces over rings, Duke Math. J. 9, 873–878 (1942).
  * [C17] Groze V. and Vasiu A., Affine structures over an arbitrary ring, Studia Univ. Babes-Bolyai Math. 25, 28–31 (1980)
  * [C18] Kaerlein G., Zur Isomorphie vektorieller Gruppen und affiner Liniengeometrien, J. Geom. 16, 1–4 (1981)
  * [C19] Keppens D., Affine planes over finite rings, a summary, Aequationes Math. 91, 979–993 (2017)
  * [C20] Lantz D., Uniqueness of Barbilian domains, J. Geom. 15, 21–27 (1981)
  * [C21] Lawrence P.A., Affine mappings in the geometries of algebras, J. Geom. 2, 115–143 (1972)
  * [C22] Leissner W., Affine Barbilian-Ebenen. I, J. Geom. 6, 31–57 (1975)
  * [C23] Leissner W., Affine Barbilian-Ebenen. II, J. Geom. 6, 105–129 (1975)
  * [C24] Leissner W., Parallelodromie–Ebenen, J. Geom. 8, 117–135 (1976)
  * [C25] Leissner W., Barbilianbereiche, in Beiträge zur Geometrischen Algebra, Birkhäuser, Basel–Stuttgart, 219–224 (1977)
  * [C26] Leissner W., Rings of stable rank 2 are Barbilian rings, Result. Math. 20, 530–537 (1991)
  * [C27] Lorimer J. W., What is a collineation of the integer plane?, Amer. Math. Mon. 103, 687–691 (1996)
  * [C28] Lüneburg H., Affine Hjelmslev–Ebenen mit transitiver Translationsgruppe, Math. Z. 79, 260–288 (1962)
  * [C29] Machala F., Affine Klingenbergsche strukturen, J. Geom. 11 , 16–34 (1978)
  * [C30] Machala F., Desarguessche affine Ebenen mit Homomorphismus, Geom. Dedicata 3, 493–509 (1975)
  * [C31] Machala F., Affine planes with homomorphism (in Czech), Kni$\breve{z}$nice, Odb. V$\breve{e}$d. Spisu Vys. U$\breve{c}$. Tech. Brn$\breve{e}$ 56, 79–84 (1975)
  * [C32] Pickert G., Taktische Konfigurationen über Quasiringen, Aequationes Math. 58, 31–40 (1999)
  * [C33] Radó F., Affine Barbilian structures, J. Geom. 14, 75–102 (1980)
  * [C34] Schleicher R., Die Eindeutigkeit der Koordinatisierung von affinen Liniengeometrien durch freie Moduln, J. Geom. 22, 143–148 (1984)
  * [C35] Schmidt S. and Steinitz R., The coordinatization of affine planes by rings, Geom. Dedicata 62, 299–317 (1996)
  * [C36] Seier W., Der kleine Satz von Desargues in affinen Hjelmslev-Ebenen, Geom. Dedicata 3, 215–219 (1974)
  * [C37] Seier W., Über Translationen in affinen Hjelmslev–Ebenen, Abh. Math. Sem. Univ. Hamburg 43, 224–228 (1975)
  * [C38] Seier W., Die Quasitranslationen desarguesscher affiner Hjelmslev-Ebenen, Math. Z. 177, 181–186 (1981)
  * [C39] Seier W., Streckungstransitive affine Hjelmslev–Ebenen, Geom. Dedicata 11, 329–336 (1981)
  * [C40] Seier W., Eine Bemerkung zum grossen Satz von Desargues in affinen Hjelmslev–Ebenen, J. Geom. 20, 181–191 (1983)
  * [C41] Sperner E., Affine Räume mit schwacher Inzidenz und zugehörige algebraische Strukturen, J. Reine und angewandte Math. 204, 205–215 (1960)
  * [C42] Vasiu A., Hjelmslev–Barbilian structures, Math., Rev. Anal. Numér. Théor. Approximation, Math. 27, 73–77 (1985)
  * [C43] Vasiu A., The coordinatisation of a class of $\mathcal{B}$–structures (in Romanian), Stud. Univ. Babes-Bolyai, Math. 31, 35–40 (1986)
  * [C44] Vasiu A., On a class of planes with neighbouring elements, Prepr., Babes-Bolyai Univ., Fac. Math., Res. Semin. 10, 59–70 (1986)

## D Metric geometry over rings

  * [D01] Bachmann F., Aufbau der Geometrie aus dem Spiegelungsbegriff, 2nd ed.Die Grundlehren der mathematischen Wissenschaften, Band 96. Springer-Verlag, Berlin-New York, 374 pp. (1973)
  * [D02] Bachmann F., Hjelmslev–Gruppen, Neudruck. Mit einem Kapitel und einem Anhang von R. Stölting. Arbeitsgemeinschaft über geometrische Fragen, Universität Kiel, Kiel, 175 pp. (1974)
  * [D03] Bachmann F., Eine neuere Entwicklung in der ebenen metrischen Geometrie, Ber. Math.-Statist. Sekt. Forsch. Graz 92–95, 22 pp. (1978)
  * [D04] Bachmann F., Ebene Spiegelungsgeometrie. Eine Vorlesung über Hjelmslev-Gruppen, Bibliographisches Institut, Mannheim, 340 pp. (1989)
  * [D05] Fischbach G., Ein Darstellungssatz für Translations-Hjelmslev-Ebenen mit Spiegelungen, J. Geom. 46, 45–54 (1993)
  * [D06] Hjelmslev J., Einleitung in die allgemeine Kongruenzlehre, 1. Mitt. Danske Vid. Selsk. mat.-fys. Medd. 8 (1929); 2. Mitt. 10 (1929); 3. Mitt. 19 (1942); 4. und 5. Mitt. 22 (1945); 6. Mitt. 25 (1949).
  * [D07] Knüppel F., Äquiforme Ebenen über kommutativen Ringen und singuläre Prä-Hjelmslev-Gruppen, Abh. Math. Semin. Univ. Hamburg 53, 229–257 (1983)
  * [D08] Knüppel F. and Kunze M., Neighbor relation and neighbor homomorphism of Hjelmslev groups, Canad. J. Math. 31, 680–699 (1979)
  * [D09] Knüppel F. and Kunze M., Reguläre Hjelmslev–Homomorphismen, Geom. Dedicata 11, 195–225 (1981)
  * [D10] Knüppel F. and Salow E., Plane elliptic geometry over rings, Pacific J. Math. 123, 337–384 (1986)
  * [D11] Kunze M., Angeordnete Hjelmslevsche Geometrie, ein Ergebnisbericht, Geom. Dedicata 10, 91–111 (1981)
  * [D12] Lingenberg R., Metric planes and metric vector spaces. Pure and Applied Mathematics, John Wiley & Sons, New York-Chichester-Brisbane, 209 pp. (1979)
  * [D13] Nolte W., Minkowskische Hjelmslevgruppen über lokalen Ringen, Result. Math. 12, 376–385 (1987)
  * [D14] Nolte W., Hjelmslevgruppen mit Nachbar-Homomorphismus, J. Geom. 38, 78–94 (1990)
  * [D15] Salow E., Singuläre Hjelmslev-Gruppen, Geom. Dedicata 1, 447–467 (1973)
  * [D16] Salow E., Einbettung von Hjelmslev-Gruppen in orthogonale Gruppen über kommutativen Ringen, Math. Z. 134, 143–170 (1973)
  * [D17] Salow E., Verallgemeinerte Halbdrehungsebenen, Geom. Dedicata 13, 67–85 (1982)
  * [D18] Schröder E., Gemeinsame Eigenschaften euklidischer, galileischer und minkowskischer Ebenen, Mitt. Math. Ges. Hamburg 10, 185–217 (1974)
  * [D19] Schröder E., Modelle ebener metrischer Ringgeometrien, Abh. Math. Sem. Univ. Hamburg 48, 139–170 (1979)
  * [D20] Schröder E., Metric geometry, chapter 17 in Handbook of Incidence Geometry: Buildings and Foundations. Edited by F. Buekenhout, Elsevier, Amsterdam, 1420 pp. (1995)
  * [D21] Stölting R., Über endliche Hjelmslev-Gruppen, Math. Z. 135, 249–255 (1973/74)
  * [D22] Stölting R., Ebene metrische Geometrien über projektiven Moduln, Abh. Math. Sem. Univ. Hamburg 50, 166–177 (1980)
  * [D23] Struve H. and Struve R., Hjelmslevgruppen, in denen sich die Punkte gegen Geraden austauschen lassen, Geom. Dedicata 13, 399–417 (1983)
  * [D24] Struve H. and Struve R., Ein spiegelungsgeometrischer Aufbau der cominkowskischen Geometrie, Abh. Math. Semin. Univ. Hamburg 54, 111–118 (1984)
  * [D25] Struve H. and Struve R., Coeuklidische Hjelmslevgruppen, J. Geom. 34, 181–186 (1989)
  * [D26] Struve R., Algebraisierung singulärer Hjelmslevgruppen, Geom. Dedicata 13, 309–323 (1982)
  * [D27] von Benda H. and Knüppel F., Hjelmslev-Gruppen über lokalen Ringen, Geom. Dedicata 5, 195–206 (1976)

## E The florescence of Hjelmslev geometry

  * [E01] Artmann B., Hjelmslev planes derived from modular lattices, Canad. J. Math. 21, 76–83 (1969)
  * [E02] Artmann B., Hjelmslev–Ebenen mit verfeinerten Nachbarschaftsrelationen, Math. Z. 112, 163–180 (1969)
  * [E03] Artmann B., Uniforme Hjelmslev–Ebenen und modulare Verbände, Math. Z. 111, 15–45 (1969)
  * [E04] Artmann B., Über die Einbettung uniformer affiner Hjelmslev–Ebenen in projektive Hjelmslev–Ebenen, Abh. Math. Semin. Univ. Hamburg 34, 127–134 (1970)
  * [E05] Artmann B., Existenz und projektive Limiten von Hjelmslev–Ebenen $n$-ter Stufe, Atti Convegno Geom. combinat. Appl. Perugia, 27–41 (1971)
  * [E06] Artmann B., Desarguessche Hjelmslev-Ebenen $n$–ter Stufe, Mitt. Math. Sem. Giessen 91, 1–19 (1971)
  * [E07] Artmann B., Geometric aspects of primary lattices, Pacific J. Math. 43, 15–25 (1972)
  * [E08] Bacon P., Strongly $n$–uniform and level $n$ Hjelmslev planes, Math. Z. 127, 1–9 (1972)
  * [E09] Bacon P., On the extension of projectively uniform affine Hjelmslev planes, Abh. Math. Semin. Univ. Hamburg 41, 185–189 (1974)
  * [E10] Brungs H. and Törner G., Embedding right chain rings in chain rings, Canad. J. Math. 30, 1079–1086 (1978)
  * [E11] Craig R., Extensions of finite projective planes. I. Uniform Hjelmslev planes, Canad. J. Math. 16, 261–266 (1964)
  * [E12] Cronheim A., Cartesian groups, formal power series and Hjelmslev planes, Arch. Math. 27, 209–220 (1976)
  * [E13] Cronheim A., Dual numbers, Witt vectors, and Hjelmslev planes, Geom. Dedicata 7, 287–302 (1978)
  * [E14] Drake D.A., Projective extensions of uniform affine Hjelmslev planes, Math. Z. 105, 196–207 (1968)
  * [E15] Drake D.A., On $n$–uniform Hjelmslev planes, J. Comb. Theory Ser. A 9, 267–288 (1970)
  * [E16] Drake D.A., The translation groups of $n$–uniform translation Hjelmlev planes, Pacific J. Math. 38, 365–375 (1971)
  * [E17] Drake D.A., The structure of $n$–uniform translation Hjelmslev planes, Trans. Amer. Math. Soc. 175, 249–282 (1973)
  * [E18] Drake D.A., Near affine Hjelmslev planes, J. Comb. Theory Ser. A 16, 34–50 (1974)
  * [E19] Drake D.A., Existence of parallelisms and projective extensions for strongly $n$–uniform near affine Hjelmslev planes, Geom. Dedicata 3, 191–214 (1974)
  * [E20] Drake D.A., Affine Hjelmslev-Ebenen mit verfeinerten Nachbarschaftsrelationen, Math. Z. 143, 15–25 (1975)
  * [E21] Drake D.A., All $n$–uniform quasitranslation Hjelmslev planes are strongly $n$–uniform, Proc. Amer. Math. Soc. 51, 494–498 (1975)
  * [E22] Drake D.A., Squeezing the accordion in a strongly $n$–uniform Hjelmslev plane, Math. Z. 185, 151–166 (1984)
  * [E23] Dugas M., Eine Kennzeichnung der endlichen desarguesschen Hjelmslev-Ebenen, Geom. Dedicata 3, 295–324 (1974)
  * [E24] Dugas M., Der Zusammenhang zwischen Hjelmslev-Ebenen und H-Verbänden, Geom. Dedicata 3, 295–324 (1974)
  * [E25] Kleinfeld E., Finite Hjelmslev planes, Illinois J. Math. 3, 403–407 (1959)
  * [E26] Lorimer J.W., The fundamental theorem of desarguesian affine Hjelmslev planes, Mitt. Math. Sem. Giessen 119, 6–14 (1975)
  * [E27] Lorimer J.W., Morphisms and the fundamental theorem of affine Hjelmslev planes, Mathematical Report 64, McMaster University, Hamilton, Ontario, Canada (1973)
  * [E28] Lorimer J. W., Structure theorems for commutative Hjelmslev rings with nilpotent radicals, C. R. Math. Rep. Acad. Sci. Canada 6, 123–127 (1984)
  * [E29] Lorimer J. W., Affine Hjelmslev rings and planes, Annals of Discr. Math. 37, 265–276 (1988)
  * [E30] Lorimer J.W. and Lane N.D., Morphisms of affine Hjelmslev planes, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 56, 880–885 (1974)
  * [E31] Lorimer J.W. and Lane N.D., Desarguesian affine Hjelmslev planes, J. für die reine und angew. Math. 1, 336–352 (1975)
  * [E32] Machala F., Über projektive Erweiterung affiner Klingenbergscher Ebenen, Czech. Math. J. 29, 116–129 (1979)
  * [E33] Machala F., Über eine Klasse affiner Klingenbergscher Ebenen, die projektiv erweiterbar sind, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat. 19, 65–74 (1980)
  * [E34] Seier W., Zentrale und axiale Kollineationen in projektiven Hjelmslev–Ebenen, J. Geom. 17, 35–45 (1981)
  * [E35] Skornjakov, L. A., Rings chain–like from the left (in Russian) Izv. Vyssh. Uchebn. Zaved. Matematika 4, 114–117 (1966)
  * [E36] Törner G., Eine klassifizierung von Hjelmslev–ringen und Hjelmslev–Ebenen, Mitt. Math. Sem. Giessen, 107, 1–77 (1974)
  * [E37] Törner G., Über Homomorphismen projektiver Hjelmslev–Ebenen, J. Geom. 5, 1–13 (1974)
  * [E38] Törner G., Homomorphismen von affinen Hjelmslev–Ebenen, Math. Z. 141, 159–167 (1975)
  * [E39] Törner G., $n$-uniforme projektive Hjelmslev-Ebenen sind stark $n$-uniform, Geom. Dedicata 6, 291–295 (1977)
  * [E40] Törner G., Über den Stufenaufbau von Hjelmslev–Ebenen, Mitt. Math. Sem. Giessen, 126, 1–43 (1977)
  * [E41] Törner G., $(r^{n-1},r)$ Hjelmslev–Ebenen des Typs $n$, Math. Z. 154, 189–201 (1977)
  * [E42] Törner G., Über ein Problem von Klingenberg, Arch. Math. 28, 253–254 (1977)
  * [E43] Törner G., Faktorisierungen von Epimorphismen projektiver Ebenen, Geom. Dedicata 18, 281–291 (1985)

## F The continuation of the Hjelmslev epoch

  * [F01] Al-Khamees Y., The enumeration of finite principal completely primary rings, Abh. Math. Sem. Univ. Hamburg 51, 226–231 (1981)
  * [F02] Civolani N., Hyperfree extensions of partial Klingenberg planes, Geom. Dedicata 9, 467–475 (1980)
  * [F03] Clark W.E. and Liang J., Enumeration of finite commutative chain rings, J. Algebra 27, 445–453 (1973)
  * [F04] Clark W.E. and Drake D.A., Finite chain rings, Abh. Math. Sem. Univ. Hamburg 39, 147–153 (1973)
  * [F05] Drake D.A., Nonexistence results for finite Hjelmslev planes, Abh. Math. Sem. Univ. Hamburg 40, 100–110 (1974)
  * [F06] Drake D.A., More new integer pairs for finite Hjelmslev planes, Illinois J. Math. 19, 618–627 (1975)
  * [F07] Drake D.A., Charakterisierungen der Hjelmslev-Ebenen mit Invarianten (4,2), Arch. Math. 27, 436–440 (1976)
  * [F08] Drake D.A., Constructions of Hjelmslev planes, J. Geom. 10, 179–193 (1977)
  * [F09] Drake D.A., The use of auxiliary sets of matrices in the construction of Hjelmslev and Klingenberg structures, Lecture Notes in Pure and Appl. Math. 82, 129–153 (1983)
  * [F10] Drake D.A. and Hale M., Group constructible $(t,k)$–nets and Hjelmslev planes, J. Algebra 48, 301–331 (1977)
  * [F11] Drake D.A. and Jungnickel D., Klingenberg structures and partial designs I: Congruence relations and solutions, J. Stat. Plann. Inference 1, 265–287 (1977)
  * [F12] Drake D.A. and Jungnickel D., Klingenberg structures and partial designs. II: Regularity and uniformity, Pacific J. Math. 76, 389–415 (1978)
  * [F13] Drake D.A. and Jungnickel D., Das Existenzproblem für projektive (8,5)-Hjelmslevebenen, Abh. Math. Sem. Univ. Hamburg 50, 118–126 (1980)
  * [F14] Drake D.A. and Jungnickel D., Finite Hjelmslev planes and Klingenberg epimorphisms, in: Rings and geometry, Proc. NATO Adv. Study Inst., Istanbul/Turkey 1984, NATO ASI Ser., Ser. C 160, 153–231 (1985)
  * [F15] Drake D.A. and Lenz H., Finite Klingenberg planes, Abh. Math. Sem. Univ. Hamburg 44, 70–83 (1975)
  * [F16] Drake D.A. and Lenz H., Finite Hjelmslev planes with new integer invariants, Bull. Amer. Math. Soc. 82, 265–267 (1976)
  * [F17] Drake D.A. and Sane S., Maximal intersecting families of finite sets and $n$–uniform Hjelmslev planes, Proc. Amer. Math. Soc. 86, 358–362 (1982)
  * [F18] Drake D.A. and Sane S., Auxiliary sets of matrices with new step parameter sequences, Linear Algebra Appl. 46, 131–153 (1982)
  * [F19] Drake D.A. and Shult E., Construction of Hjelmslev planes from $(t,k)$–nets, Geom. Dedicata 5, 377–392 (1976)
  * [F20] Drake D.A. and Törner G., Die Invarianten einer Klasse projektiver Hjelmslev–Ebenen, J. Geom. 7, 157–174 (1976)
  * [F21] Hale M. and Jungnickel D., A generalization of Singer’s theorem, Proc. Amer. Math. Soc. 71, 280–284 (1978)
  * [F22] Jungnickel D., Verallgemeinerte Klingenberg–Ebenen, Mitt. Math. Sem. Giessen 120, 1–10 (1976)
  * [F23] Jungnickel D., Hjelmslevebenen mit regulärer abelscher Kollineationsgruppe, Beiträge zur geometrischen Algebra (Proc. Sympos. Duisburg 1977), 157–165 (1977)
  * [F24] Jungnickel D., Regular Hjelmslev planes. II, Trans. Amer. Math. Soc. 241, 321–330 (1978)
  * [F25] Jungnickel D., On the congruence relations of regular Klingenberg structures, J. Combin. Inform. System Sci. 3, 49–57 (1978)
  * [F26] Jungnickel D., Regular Hjelmslev planes, J. Comb. Theory Ser. A 26, 20–37 (1979)
  * [F27] Jungnickel D., On an assertion of Dembowski, J. Geom. 12, 168–174 (1979)
  * [F28] Jungnickel D., Construction of regular proper CK–planes, J. Combin. Inform. System Sci. 4, 14–18 (1979)
  * [F29] Jungnickel D., On balanced regular Hjelmslev planes, Geom. Dedicata 8, 445–462 (1979)
  * [F30] Jungnickel D., Some new combinatorial results on finite Klingenberg structures, Utilitas Math. 16, 249–269 (1979)
  * [F31] Jungnickel D., A class of uniform Klingenberg matrices, Ars Combin. 10, 91–94 (1980)
  * [F32] Limaye B.V. and Sane S., On partial designs and $n$–uniform projective Hjelmslev planes, J. Combin. Inform. System Sci. 3, 223–227 (1978)
  * [F33] Neumaier A., Nichtkommutative Hjelmslev-Ringe, Festband für H. Lenz, Freie Universität Berlin, 200–213 (1976)
  * [F34] Sane S., Some new invariant pairs $(t,3)$ for projective Hjelmslev planes, J. Geom. 15, 64–73 (1981)
  * [F35] Sane S., New integer pairs for Hjelmslev planes, Geom. Dedicata 10, 35–48 (1981)
  * [F36] Sane S., On class-regular projective Hjelmslev planes, in: Finite geometries and designs, London Math. Soc. Lecture Note Ser. 49, 332–336 (1981)
  * [F37] Sane S., On the theorems of Drake and Lenz, Aequationes Math. 23, 223–232 (1981)
  * [F38] Sane S. and Singhi N., On the structure of a finite projective Klingenberg plane, Congr. Numer. 33, 285–292 (1981)
  * [F39] Sedlar V., Incidence matrices of finite projective uniform Hjelmslev planes (in Czech) Sb. Praci Ped. Fak. v Ostrave Ser. A 17, 25–38 (1982)

## G The coordinatization of Klingenberg and Hjelmslev planes

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  * [G05] Bacon P., Desarguesian Klingenberg planes, Trans. Amer. Math. Soc. 241, 343–355 (1978)
  * [G06] Baker, C., Moulton affine Hjelmslev planes Canad. Math. Bull. 21, 135–142 (1978)
  * [G07] Baker, C., Lane N. and Lorimer J.W., Local alternative rings and finite alternative right chain rings, C. R. Math. Rep. Acad. Sci. Canada 12, 53–58 (1990)
  * [G08] Baker, C., Lane N. and Lorimer J.W., An affine characterization of Moufang projective Klingenberg planes, Results Math. 17, 27–36 (1990)
  * [G09] Baker, C., Lane N. and Lorimer J.W., The Artin-Zorn theorem for finite punctually cohesive projective Klingenberg planes, Ars Comb. Ser. B 29, 143–149 (1990)
  * [G10] Baker, C., Lane N. and Lorimer J.W., A coordinatization for Moufang Klingenberg planes, Simon Stevin 65, 3–22 (1991)
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  * [G12] Blunck A., Projectivities in Moufang-Klingenberg planes, Geom. Dedicata 40, 341–359 (1991)
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  * [G19] Çelik B., Akpinar A. and Çiftçi S., On harmonicity in some Moufang-Klingenberg planes, Turk. J. Math. 34, 249–260 (2010)
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  * [G21] Cyganova V. K., Ternary rings of affine Hjelmslev planes (in Russian), Smolensk. Gos. Ped. Inst. Uchen. Zap. 18, 44–69 (1967)
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  * [G27] Cyganova V. K., The second minor Pappos theorem and its algebraic equivalent in Hjelmslev affine planes (in Russian), Vestsi Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk 1, 107–109 (1984)
  * [G28] Drake D.A., Coordinatization of $H$-planes by $H$-modules, Math. Z. 115, 79–103 (1970)
  * [G29] Dugas M., Charakterisierungen endlicher Desarguesscher uniformer Hjelmslev-Ebenen, Geom. Dedicata 3, 295–324 (1974)
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  * [G31] Dugas M., Verallgemeinerte André–Ebenen mit Epimorphismen auf Hjelmslev–Ebenen, Geom. Dedicata 8, 105–123 (1979)
  * [G32] Eliseev, E.M., Desarguesian theorems and collineations of projective Hjelmslev planes, (in Russian) Geometry of incidence structures and differential equations, Collect. sci. Artic., Smolensk, 30–40 (1981)
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  * [G34] Emel’chenkov E.P., The PH–ternary of a projective Hjelmslev plane (in Russian), Smolenk. Gos. Ped. Inst. Uchen. Zap. 4, 93–101 (1973)
  * [G35] Emel’chenkov E.P., Homotheties of AH–planes and H–ternaries (in Russian), Gos. Ped. Inst., Leningrad, Geom. Topol. 2, 89–93 (1974)
  * [G36] Emel’chenkov E.P., On $(\Pi,l)$–collineations of AH–planes (in Russian), in Modern Geometry, Gos. Ped. Inst., Leningrad, 58–60 (1978)
  * [G37] Hall M., Projective planes, Trans. Amer. Math. Soc. 54, 229–277 (1943)
  * [G38] Hughes D.R., Planar division neo-rings, Trans. Amer. Math. Soc. 80, 502–527 (1955)
  * [G39] Hughes D.R. and Piper F.C, Projective planes, Springer–Verlag, Berlin, 291 pp. (1973)
  * [G40] Jukl M., Desargues theorem for Klingenberg projective plane over certain local ring, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 36, 33–39 (1997)
  * [G41] Keppens D., Coordinatization of projective Klingenberg planes. I: Introduction of coordinates and planar sexternary rings, Simon Stevin 62, 63–90 (1988)
  * [G42] Keppens D., Coordinatization of projective Klingenberg planes. II: Connections between geometric properties of a PK-plane and algebraic properties of a coordinatizing PSR, Simon Stevin 62, 163–188 (1988)
  * [G43] Keppens D., Coordinatization of projective Klingenberg planes. III: Construction of planar sexternary rings and examples, Simon Stevin 63, 117–140 (1989)
  * [G44] Kolb E., Projective Klingenberg planes over nearrings, J. Geom. 46, 82–91 (1993)
  * [G45] Kolb E., Hjelmslev planes over nearrings, Discrete Math. 155, 147–155 (1996)
  * [G46] Lorimer J.W., Coordinate theorems for affine Hjelmslev planes, Ann. Mat. Pura Appl. 105, 171–190 (1975)
  * [G47] Machala F., Erweiterte lokale Ternärringe, Czech. Math. J. 27 , 560–572 (1977)
  * [G48] Machala F., Koordinatisation projektiver Ebenen mit Homomorphismus, Czech. Math. J. 27 , 573–590 (1977)
  * [G49] Machala F., Koordinatisation affiner Ebenen mit Homomorphismus, Math. Slovaca 27, 181–193 (1977)
  * [G50] Machala F., Projektive Ebenen mit Homomorphismus und erweiterte lokale Ternärringe, Math. Slovaca 29, 227–237 (1979)
  * [G51] Machala F., Epimorphismen von lokalen Ternärringen, Czech. Math. J. 33, 70–75 (1983)
  * [G52] Machala F., Biternärringe und affine lokale Ternärringe, Acta. Univ. Palack. Olomuc. Fac. Rerum. Natur. Math. 27 , 25–37 (1988)
  * [G53] Mäurer H. and Nolte W., A characterization of Pappian affine Hjelmslev planes, Combinatorics ’86 (Trento, 1986), Ann. Discrete Math. 37, 281–291 (1988)
  * [G54] Nolte W., Pappussche affine Klingenbergebenen, J. Geom. 52, 152–158 (1995)
  * [G55] Pickert G., Projektive Ebenen, Springer–Verlag, Berlin, second edition, 388 pp. (1975)
  * [G56] Shatokhin N. L., Homomorphisms of H-planes and PH-ternars (in Russian), Geometry of incidence structures and differential equations, Collect. sci. Artic., Smolensk, 81–91 (1981)
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  * [G58] Skornyakov L.A., Natural domains of Veblen-Wedderburn projective planes (in Russian), Izvestiya Akad. Nauk SSSR. Ser. Mat. 13, 447–472 (1949); english translation in Amer. Math. Soc. Translation 58, 37 pp. (1951)
  * [G59] Skornyakov L.A., Projective planes (in Russian), Uspehi Matem. Nauk (N.S.) 6, 112–154 (1951); english translation in Amer. Math. Soc. Translation 99, 58 pp. (1953)

## H Order and topology in ring geometry

  * [H01] Baker C., Ordered affine Hjelmslev planes, J. Geom. 23, 1–13 (1984)
  * [H02] Baker C., Preordered uniform Hjelmslev planes, J. Geom. 24, 14–17 (1985)
  * [H03] Baker C. and Lorimer J.W., Coordinate rings of topological Klingenberg planes. I: The affine perspective, Geom. Dedicata 58, 101–116 (1995)
  * [H04] Baker C., Lane N.D. and Lorimer J.W., Order and topology in projective Hjelmslev planes, J. Geom. 19, 8–42 (1982)
  * [H05] Baker C., Lane N.D. and Lorimer J.W., A construction for topological non–desarguesian affine Hjelmslev planes, Arch. Math. 50, 83–92 (1988)
  * [H06] Baker C., Lane N.D., Laxton J.A. and Lorimer J.W., Preordered affine Hjelmslev planes, J. Geom. 23, 14–44 (1984)
  * [H07] Lorimer J.W., Topological Hjelmslev planes, Geom. Dedicata 7, 185–207 (1978)
  * [H08] Lorimer J.W., Connectedness in topological Hjelmslev planes, Ann. Mat. Pura Appl. 118, 199–216 (1978)
  * [H09] Lorimer J.W., Locally compact Hjelmslev planes, C. R. Math. Rep. Acad. Sci. Canada 1, 309–314 (1978/79)
  * [H10] Lorimer J.W., Locally compact Desarguesian Hjelmslev planes of level $n$, C. R. Math. Rep. Acad. Sci. Canada 2, 141–145 (1980)
  * [H11] Lorimer J.W., Locally compact Hjelmslev planes and rings, Canad. J. Math. 33, 988–1021 (1981)
  * [H12] Lorimer J.W., Dual numbers and topological Hjelmslev planes, Canad. Math. Bull. 26, 297–302 (1983)
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  * [H15] Lorimer J.W., The classification of compact punctally cohesive Desarguesian projective Klingenberg planes, Geom. Dedicata 36, 347–358 (1990)
  * [H16] Lorimer J.W., Topological characterizations of finite desarguesian projective Hjelmslev planes, Ars Comb. Ser. A 29, 247–254 (1990)
  * [H17] Lorimer J.W., The classification of compact right chain rings, Forum Math. 4, 335–347 (1992)
  * [H18] Machala F., Angeordnete affine Klingenbergsche Ebenen, Czech. Math. J. 30, 341—356 (1980)
  * [H19] Machala F., Angeordnete affine lokale Ternärringe und angeordnete affine Klingenbergsche Ebenen, Czech. Math. J. 30, 556–568 (1980)
  * [H20] Machala F., Fastgeordnete und geordnete affine Klingenbergsche Ebenen, Čas. Pěst. Mat. 106, 138–155 (1981)
  * [H21] Machala F., Fastgeordnete und geordnete lokale Ringe und ihre geometrische Anwendung, Čas. Pěst. Mat. 106, 269—278 (1981)
  * [H22] Machala F., Über angeordnete affine Klingenbergsche Ebenen, die sich in projektive Klingenbergsche Ebenen einbetten lassen, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 21, 9–31 (1982)
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  * [H25] Machala F., Fastgeordnete affine lokale Ternärringe und pregeordnete Biternärringe, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 28, 11–25 (1989)
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  * [H30] Thomas L., Ordered desarguesian affine Hjelmslev planes, Canad. Math. Bull. 21, 229–235 (1978)

## I The revival of ring geometry (Veldkamp, Faulkner)

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  * [I02] Faulkner J.R., Coordinatization of Moufang–Veldkamp planes, Geom. Dedicata 14, 189–201 (1983)
  * [I03] Faulkner J.R., Barbilian planes, Geom. Dedicata 30, 125–181 (1989)
  * [I04] Faulkner J.R., A geometric construction of Moufang planes, Geom. Dedicata 29, 133–140 (1989)
  * [I05] Faulkner J.R., Current results on Barbilian planes, Sci. Bull., Politeh. Univ. Buchar., Ser. A 55, 146–152 (1993)
  * [I06] Faulkner J.R., Projective remoteness planes, Geom. Dedicata 60, 237–275 (1996)
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  * [I11] Hanssens G. and Van Maldeghem H., A note on near–Barbilian planes, Geom. Dedicata 29, 233–235 (1989)
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  * [I16] Moore D’Ortona C., Homomorphisms of projective remoteness planes, Geom. Dedicata 72, 111–122 (1998)
  * [I17] Péntek K., The minimal projective ring planes (in Hungarian), Berzsenyi Dániel Tanárk. Föisk. Tud. Közl., Termtud. 8, 33–70 (1992)
  * [I18] Péntek K., A generalization of the projective Klingenberg plane, Berzsenyi Dániel Tanárk. Föisk. Tud. Közl., Termtud. 9, 19–42 (1994)
  * [I19] Péntek K., The $\Delta^{2}$–configuration and the direct decomposition of projective Veldkamp planes (in Hungarian), Berzsenyi Dániel Tanárk. Föisk. Tud. Közl., Termtud. 10, 27–37 (1996)
  * [I20] Péntek K., On the direct decomposition of pappian projective Veldkamp planes, Publ. Math. 53, 347–365 (1998)
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  * [I22] Spanicciati R., Near–Barbilian planes, Geom. Dedicata 24, 311–318 (1987)