# 50 years of Finite Geometry,
the “geometries over finite rings” part

Dirk Keppens

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

Whereas for a substantial part, “Finite Geometry” during the past 50 years has
focussed on geometries over finite fields, geometries over finite rings that
are not division rings have got less attention. Nevertheless, several
important classes of finite rings give rise to interesting geometries.
In this paper we bring together some results, scattered over the literature,
concerning finite rings and plane projective geometry over such rings. It
doesn’t contain new material, but by collecting stuff in one place, we hope to
stimulate further research in this area for at least another 50 years of
Finite Geometry.

Keywords: Ring geometry, finite geometry, finite ring, projective plane

AMS Classification: 51C05,51E26,13M05,16P10,16Y30,16Y60

##  1 Introduction

Geometries over rings that are not division rings have been studied for a long
time. The first systematic study was done by Dan Barbilian [11], besides a
mathematician also one of the greatest Romanian poets (with pseudonym Ion
Barbu). He introduced plane projective geometries over a class of associative
rings with unit, called Z–rings (abbreviation for Zweiseitig singuläre Ringe)
which today are also known as Dedekind-finite rings. These are rings with the
property that $ab=1$ implies $ba=1$ and they include of course all commutative
rings but also all finite rings (even non–commutative).

Wilhelm Klingenberg introduced in [52] projective planes and 3–spaces over
local rings. A ring $R$ is local if it possesses a unique maximal right ideal
(which turns out to be the Jacobson radical $J(R)$). For a local ring $R$ the
quotient ring $R/J(R)$ is a division ring (= skewfield or field) and the
natural homomorphism of $R$ onto $\mathbb{K}=R/J(R)$ induces an epimorphism of
the plane $P_{2}(R)$ over $R$ onto the ordinary desarguesian projective plane
PG(2,$\mathbb{K}$). Nowadays planes over local rings are called (desarguesian)
Klingenberg planes (see also [7]). In the finite case such planes have the
finite projective plane PG(2,$q$) over the Galois field GF($q$) as epimorphic
image.

In three other papers [49], [50] and [51], Klingenberg studied projective
planes over local rings with some additional properties, called $H$–rings
(short for Hjelmslev rings). In these rings the left and right ideals form a
chain and the maximal ideal contains only zero divisors. If one drops that
last condition, one gets chain rings. In the finite case any chain ring is an
$H$–ring. Planes over $H$–rings are now called (desarguesian) Hjelmslev planes
after the Danish mathematician Johannes Hjelmslev (born as Johannes Petersen)
who was the first one to consider plane geometries in which two distinct lines
may have more than one point in common [41]. Among the finite $H$–rings are
the Galois rings GR($p^{nr},p^{n}$) of cardinality $p^{nr}$ and characteristic
$p^{n}$ which are natural generalizations of Galois fields.

In the early seventies another class of rings came under the attention: full
matrix rings over fields. Strongly inspired by the work of the italian “father
of Galois geometry” Beniamino Segre on geometries over finite fields (e.g.
[77]), J.A. Thas defined projective planes (and higher dimensional spaces)
over full matrix rings with elements in a field and investigated combinatorial
properties in the finite planes over the matrix rings $M_{n}(GF(q))$ of
$n\times n$–matrices over Galois fields [80]. We will refer to these planes
further as Thas planes.

In the eighties F.D. Veldkamp was very productive in the area of projective
ring planes and their generalizations. He gave in [84] and [86] an axiomatic
description of projective planes and higher dimensional geometries over the
large class of rings of stable rank 2, a notion coming from algebraic
$K$–theory.
A ring $R$ has stable rank 2 if for any $a,b\in R$ with $Ra+Rb=R$ there exists
$r\in R$ such that $a+rb$ is a unit. The class of rings of stable rank 2
includes the class of semilocal rings (hence also all finite rings, local
rings, chain rings, $H$–rings and matrix rings over a division ring) and a
ring of stable rank 2 is always Dedekind–finite (hence a Z–ring in the sense
of Barbilian). Projective planes over rings of stable rank 2 are called
(desarguesian) Veldkamp planes. Among these are Klingenberg planes, Hjelmslev
planes, Thas planes and also the projective planes over semiprimary rings
(i.e. rings with nilpotent Jacobson radical and with $R/J(R)$ semisimple)
treated by Bingen in [13].

In almost all papers on projective geometry over rings no special attention is
paid to the finite case. Mostly, theorems deal with rings in general (with no
specification for finite or infinite). In this paper we restrict ourselves to
the finite case. First we bring together some results on finite rings with
special attention for local rings. Then we have a closer look at projective
plane geometries over finite rings. In the last section we deal with some
generalizations of rings (semirings, nearrings and alternative rings) and
projective plane geometries over such algebraic structures.

##  2 Finite rings

In this section the word “ring” always refers to an associative ring with unit
$1\not=0$, but with multiplication not necessarily commutative.

Finite fields or Galois fields are well-known algebraic structures. Finite
fields of order $q$ only exist if $q$ is a prime power ($q=p^{r}$) and for
each such $q$ their is a unique (up to isomorphism) field of that order which
is denoted by $\mathbb{F}_{q}$ or by GF($q$). The prime number $p$ is the
characteristic of the field.

It is natural to look at generalizations of finite fields to finite rings, but
the situation is much more complicated. First there exist finite
non–commutative rings unlike the situation in finite division rings where the
famous theorem of Wedderburn forces any finite skewfield to be a field. Also
the order of a finite ring doesn’t uniquely determine that ring (for example
there are four non–isomorphic rings of order four, including one field). A
complete classification of finite rings seems to be a “mission impossible”
(even if one restricts to the commutative case).
The paper of Raghavendran [72] on rings of prime power order was the starting
point for the study of the structure of finite rings. Also the work of Wilson
[88] and [89] was of great importance. A recent survey on results obtained so
far with an extensive bibliography can be found in Nechaev [68].

Local rings, first defined by Krull in [56], play a central role in the
structure theory of (finite) rings. Recall that a ring $R$ is called local if
it possesses a unique maximal right ideal (or equivalently a unique maximal
left ideal). This is stronger than asking that $R$ has a unique maximal
two–sided ideal (e.g. the ring $M_{n}(\mathbb{Z}_{p^{n}})$ of $n\times
n$–matrices over $\mathbb{Z}/p^{n}\mathbb{Z}$ has a unique maximal two–sided
ideal but is not local). The unique maximal right or left ideal in a local
ring turns out to be the Jacobson radical $J(R)$. Other characterizations of
local rings are possible. E.g. $R$ is local iff the set of non–units forms a
right (or left) proper ideal in $R$. Also $R$ is local if and only if for all
$r\in R$ either $r$ or $1-r$ is invertible. Finally $R$ is local iff $R/J(R)$
is a division ring. Other characterizations in terms of zero divisors are
given in [76]. In the finite case one can say even more: $R$ is local iff
$R\setminus J(R)$ is the set of units of $R$ or equivalently $J(R)$ is the set
of nilpotent elements of $R$.

The following theorem gives parameters for finite local rings.

######  Theorem 2.1.

(Raghavendran [72]) Let $R$ be a finite local ring. Then there exist unique
numbers $p$, $n$, $r$ and $k$ such that $|R|=p^{nr}$, $|J(R)|=p^{(n-1)r}$ and
the characteristic of $R$ is $p^{k}$ with $1\leq k\leq n$. The number $p^{r}$
is the order of the Galois field $R/J(R)$ and the number $n$ is the index of
nilpotency of $J(R)$. If $k=n$ then $R$ is commutative.

There is also a more recent result which conversely characterizes local rings
among finite rings just by a couple of parameters.

######  Theorem 2.2.

(Behboodi and Beyranvand [12], González [36]) Let $R$ be a finite ring and let
$Z(R)$ be the set of zero–divisors of $R$. Then $R$ is local if and only if
$|R|=p^{m}$ and $|Z(R)|=p^{n}$ for some prime number $p$ and integers $1\leq
n<m$. Moreover, when $R$ is local with these parameters, then the order of
$R/J(R)=R/Z(R)=p^{r}$ with $r=m-n$.

The structure of commutative finite local rings was first studied by Ganske
and McDonald [33]. Classification theorems are proved for fixed orders or
fixed characteristic in [16], [17], [24] and [72]. In the non–commutative case
Wirt [90] has contributed to the theory.

By the following important structure theorem the classification problem of
commutative finite rings can be reduced to that of finite local rings.

######  Theorem 2.3.

(McDonald [67]) Let $R$ be a finite commutative ring. Then $R$ decomposes (up
to order of summands) uniquely as a direct sum of finite local rings.

Another decomposition theorem, also valid in the non–commutative case, shows
once more the importance of finite local rings.

######  Theorem 2.4.

(McDonald [67] and Wirt [90]) Let $R$ be a finite ring. Then $R$ decomposes as
$S+N$ with $S$ a direct sum of full matrix rings over finite local rings and
$N$ a subring of the Jacobson radical $J(R)$.

Next we look at principal ideal rings. A ring is called a right principal
ideal ring if any right ideal $I$ is right principal, i.e. generated by one
element ($I=aR$). Similar definition for left principal ideal ring. If a ring
is a left and right principal ideal ring, it is called a principal ideal ring
(PIR). A right principal ideal ring is always a right noetherian ring, since
any right ideal is finitely generated. It is also a right Bézout ring since
any finitely generated right ideal is principal. In fact the right PIR’s are
just the rings which are both right noetherian and right Bézout. Similar
results are true for left principal ideal rings and PIR’s.
The structure of finite principal ideal rings was first studied by Fisher in
[32]. For finite rings the notions of left PIR, right PIR and PIR are
equivalent (see [68]).

Another important class of rings are the chain rings. A ring is called a right
chain ring if for any $a$ and $b$ in $R$ either $a\in bR$ or $b\in aR$. For a
right chain ring the lattice of right ideals is totally ordered by inclusion
and it follows that $R$ is a local ring. Analogous definitions and results for
left chain rings. A ring which is a left and right chain ring is called a
chain ring. In the infinite case there are examples of right chain rings which
are not chain rings (see [63], [79] and [15]), but in the finite case there is
a left–right equivalence. Every ideal of a chain ring is a power of the unique
maximal ideal. A (left or right) chain ring with the additional property that
any non–unit is a two–sided zero divisor is called a (left or right) $H$–ring
or affine Hjelmslev ring (two–sided chain rings are also known as projective
Hjelmslev rings). Finite chain rings are always left and right $H$–rings. For
a comprehensive study of $H$–rings linked to the behaviour of ideals, we refer
to [81].

The following theorem shows that finite chain rings are nothing but finite
local principal ideal rings!

######  Theorem 2.5.

(Clarke and Drake [18] and Lorimer [62]) Let $R$ be a finite ring. Then the
following conditions are equivalent: (a) $R$ is a local PIR (b) $R$ is a local
ring with principal maximal ideal (b) $R$ is a left chain ring (c) $R$ is a
right chain ring (d) $R$ is a chain ring

A valuation ring in a division ring $\mathbb{D}$ is a proper subring $R$ with
the property that $x$ or $x^{-1}$ $\in R$ for each nonzero $x\in\mathbb{D}$. A
ring is a valuation ring if and only if it is a left and right chain domain
(i.e. a chain ring without zero divisors) (for a proof, see [60]. Since any
finite domain is a finite field, there do not exist finite valuation rings.

A ring $R$ is called E–ring if and only if it possesses an ideal $I$ so that
all ideals of $R$ are of the form $I^{n}$. In the infinite case $E$–rings can
be characterized as $H$–rings with nilpotent radical and also as proper
homomorphic images of discrete valuation rings (see [4], [5], [60] and [61]).
In the finite case the notions of $H$–ring and $E$–ring coincide.

The simplest and most investigated finite chain rings are the Galois rings,
first defined by Krull [56] as “Grundringe” and later rediscovered by Janusz
[46] and Raghavendran [72]. A Galois ring is a commutative local PIR such that
$J(R)=(p)$ with $p=1+1+\ldots+1$ ($p$ terms) for some prime $p$. These rings
are very close to Galois fields. In the past ten years, finite chain rings and
in particular Galois rings got a lot of attention in connection with coding
theory (see e.g. [42] and [43]).

As for Galois fields one has

######  Theorem 2.6.

(Raghavendran [72] and McDonald [67]) For any prime $p\in\mathbb{N}$ and for
any $n,r\in\mathbb{N}$ there exists a unique (up to isomorphism) Galois ring
$R$ consisting of $q^{n}$ (with $q=p^{r}$) elements and with characteristic
$p^{n}$

The unique Galois ring in the preceding theorem is denoted by
GR($q^{n},p^{n}$) (or sometimes also by GR($p^{n},r$)). For $p=q$ we have
GR($p^{n},p^{n}$) = $\mathbb{Z}_{p^{n}}$, the ring of integers module $p^{n}$,
and for $n=1$ we obtain the Galois field GF($q$)=GR($q,p$). All Galois rings
can be constructed in the form $R=\mathbb{Z}_{p^{n}}[x]/(f(x))$ where $f(x)$
is a monic polynomial of degree $r$ which is irreducible modulo $p$ and hence
GR($q^{n},p^{n})$, $q=p^{r}$, can be seen as Galois extensions of degree $r$
of its subring $\mathbb{Z}_{p^{n}}$.

The properties of Galois rings are well known, e.g. the structure of the group
of units, the automorphism group, the possible subrings etc. Many results can
be found in [14]. The classification of all chain rings is still an open
problem but partial results are known. Galois rings occur in the construction
of finite chain rings as can be seen from next theorem.

######  Theorem 2.7.

(Clark and Liang [19], Wirt [90], Neumaier [69]) Let $R$ be a finite chain
ring with parameters $p,n,r$ and $k$ as in theorem 2.1. Then there exist
integers $t$ and $s$ such that $R$ is isomorphic to
$S[x,\sigma]/(g(x),p^{k-1}x^{t})$ with $S$ = GR($q^{k},p^{k}$) and
$S[x,\sigma]$ the Ore–skew polynomial ring over $S$, i.e. with usual addition
and the multiplication $xa=\sigma(a)x$ for $\sigma\in Aut\;S$, and with
$g(x)\in$ $S[x,\sigma]$ an Eisenstein polynomial of degree $s$,
$g(x)=x^{s}-p(a_{0}+a_{1}x+\ldots+a_{s-1}x^{s-1})$ with $a_{0}$ a unit in $S$
and $n=(k-1)s+t$ ($1\leq t\leq s\leq n$)

The integer $s$ in the theorem above is called the ramification index of $R$.
It is the smallest integer such that the ideal $(p)$ is equal to $J(R)^{s}$.

For a given set of parameters $p,n,r,k,s$ and $t$ one could ask for the number
of non–isomorphic finite chain rings. In general this problem is still open,
but partial results are known (see e.g. [1], [3], [74], [68] ). In some cases
the parameters determine completely the ring:

######  Theorem 2.8.

(Clark and Liang [19] and Arkhipov [3]) Let $R$ be a finite chain ring with
parameters $p,n,r,k,t$ and $s$ as in theorem 2.7.

(a) If $k=1$ (hence $R$ has minimal characteristic $p$), then $R$ is uniquely
determined (up to isomorphism) and $R\cong$ GF($q$)$[x,\sigma]/(x^{n})$ (a
truncated skew polynomial ring)

(b) If $k=n$ (hence $R$ has maximal characteristic $p^{r}$), then $R$ is
uniquely determined (up to isomorphism) and $R\cong$ GR($q^{n},p^{n}$) (always
commutative)

Some more results are known for finite chain rings with characteristic $p^{k}$
with $1<k<n$ (see [44]). In [91] still another description of finite
(commutative) chain rings is given as certain homomorphic images of the
polynomial ring $\mathbb{Z}_{p^{r}}[x,y]$.

An important special subclass of chain rings is the one for which the Jacobson
radical $J$ of $R$ has index of nilpotency $2$, so $J^{2}=0$. In this case
$n=2$ and the two cases in theorem 2.8 are the only possible ones. Finite
chain rings with $J\not=0$ and $J^{2}=0$ are called uniform. The
classification of finite uniform chain rings follows from theorem 2.8 but was
also proved directly by Cronheim.

######  Theorem 2.9.

(Cronheim [25]) Every finite uniform chain ring with $R/J\cong$ GF($q$) is
either a ring of (twisted) dual numbers, or a truncated Witt ring of length 2,
over the field GF($q$).

Rings of (twisted) dual numbers are the rings $\mathbb{D}(q,\sigma)$ =
GF($q$)$[x,\sigma]/(x^{2})$ (twisted for $\sigma\not=1$) (corresponding to
case ($a$) with $n=2$ in theorem 2.8).
$\mathbb{D}(q,\sigma)$ can also be represented as the subring of matrices
$\left(\begin{array}[]{ll}a&b\\\ 0&a^{\sigma}\end{array}\right)$ in the full
matrix ring $M_{2}(q)$ of $2\times 2$ matrices with elements in GF($q$).

$W_{2}(q)$, the truncated Witt ring of length 2 over GF($q$) is defined on the
set GF($q$) $\times$ GF($q$), $q=p^{k}$, as follows:

addition:
$(x_{0},x_{1})+(y_{0},y_{1})=(x_{0}+y_{0},x_{1}+y_{1}+\dfrac{x_{0}^{p}+y_{0}^{p}-(x_{0}+y_{0})^{p}}{p})$

and multiplication:
$(x_{0},x_{1})\cdot(y_{0},y_{1})=(x_{0}y_{0},x_{0}^{p}y_{1}+x_{1}y_{0}^{p})$

It can be proved that $W_{2}(q)$ is isomorphic to the Galois ring
GR($q^{2},p^{2}$) (this is case ($b$) with $n=2$ in theorem 2.8).

##  3 Finite ring planes

In this section we deal with geometries over finite rings and we restrict
ourselves to the case of plane projective geometries. The projective line,
higher dimensional projective geometries, affine and metric geometries, planar
circle geometries (Benz–planes), chain geometries and polar geometries over
finite rings will not be considered here.

A big part of finite geometry, called Galois geometry, is related to finite
fields, see e.g. the work of Hirschfeld [40]. Since the pioneering work of B.
Segre, the finite desarguesian (pappian) projective plane PG($2,q$) and its
interesting point sets (arcs, ovals, blocking sets, unitals, $\ldots$) have
been studied extensively. For planes over finite rings still a lot of work has
to be done. As already mentioned in the introductory section, plane geometries
over some important classes of rings have been defined in a suitable way,
starting somewhere in 1940 by Barbilian (some isolated cases over particular
rings were even known longer ago).

Before we look at planes over finite rings, we first recall the definition of
a projective plane over an arbitrary ring (not necessarily finite).

Let $R$ be an arbitrary ring (associative and with unit element). Denote the
set of twosided invertible elements of $R$ by $R^{\star}$. Following [11],
[26] or [55], we can construct a plane projective geometry PG($2,R$) over $R$
as follows:

points are the left unimodular triples $(x,y,z)\in R\times R\times R$ up to a
right scalar in $R^{\star}$ (where $(x,y,z)$ left unimodular means that there
exist $a,b,c\in R$ such that $ax+by+cz=1$ or equivalently $Rx+Ry+Rz=R$)

lines are the right unimodular triples $[u,v,w]\in R\times R\times R$ up to a
left scalar in $R^{\star}$ (where $[u,v,w]$ right unimodular means that there
exist $a,b,c\in R$ such that $ua+vb+wc=1$ or equivalently $xR+yR+zR=R$)

incidence I (between points and lines) is defined as follows: $(x,y,z)$ I
$[u,v,w]$ if and only if $ux+vy+wz=0$

neighborship $\sim$ (between points and lines) is defined by:
$(x,y,z)\sim[u,v,w]$ if and only if $ux+vy+wz\in R\setminus R^{\star}$
It is clear that incidence always implies neighborship, so $p$ I $L$ implies
$p\sim L$ for any point $p$ and any line $L$.

The incidence structure (with neighbor relation) obtained in this way is
called right projective plane over $R$. In the same way one can define the
left projective plane over $R$ which is clearly isomorphic to the right
projective plane over the opposite ring $R^{\circ}$. Therefore we will drop
from now on the specification “right” or “left”.

Although the denomination “projective plane” is used here, the projective
plane over a ring (which is not a division ring) isn’t a projective plane in
the usual sense, as two distinct points may be incident with none or with more
than one line and dually.

In addition to the neighbor relation for point–line pairs, also a neighbor
relation between points (between lines respectively) can be considered in
PG($2,R$): points $(x,y,z)$ and $(x^{\prime},y^{\prime},z^{\prime})$ are
neighboring iff $\left[\begin{array}[]{lll}x&y&z\\\
x^{\prime}&y^{\prime}&z^{\prime}\end{array}\right]$ cannot be extended to an
invertible $3\times 3$–matrix over $R$ (and similar for lines).

Dealing with projective planes over rings it is natural to assume that
non–neighboring elements behave the same as distinct elements in an ordinary
projective plane over a division ring. To get that situation, it is necessary
to restrict to the class of rings for which every one–sided unit is a
two–sided unit, which was first observed by Barbilian [11]. Indeed, assume
that $r$ is right–invertible (with right inverse $a$), but not left
invertible. Consider the lines $[1,0,0]$ and $[r,0,0]$ (remark that $[r,0,0]$
is right unimodular since $r\cdot a+0\cdot b+0\cdot c=1$). These lines are
distinct as otherwise there would exist a left scalar $l\in R^{\star}$ for
which $[1,0,0]=l\cdot[r,0,0]$, so $1=l\cdot r$ in contradiction with the
assumption that $r$ has not a left inverse. Now these two distinct lines are
incident with the non–neighboring points $(0,1,0)$ and $(0,0,1)$.

The restriction to rings in which any left (or right) invertible element is a
two–sided unit (or equivalently $a\cdot b=1$ implies $b\cdot a=1$), the
so–called Dedekind–finite rings, only comes up when one deals with infinite
rings. In the finite case (but also for many important classes of infinite
rings) invertible elements are always two–sided invertible. In this context it
is also interesting to mention that all reversible rings (i.e. $a\cdot b=0$
implies $b\cdot a=0$) are Dedekind–finite.

Next we are interested in the connection between properties of the ring $R$
and the projective plane PG($2,R$). Most of the following results are
reformulations for the finite case of theorems that can be found in Veldkamp
[84] and [85] .

In the projective plane over a Dedekind–finite ring, there is a unique line
incident with two given non–neighboring points (and dually). One might ask
whether the neighbor relation is completely determined by the incidence
relation in the sense that two points are non–neighboring if and only if there
is unique line incident with them (and dually for lines). This is not always
the case, but it does if every non–invertible element in $R$ is a right and
left zero–divisor.

A (left or right) artinian ring is a ring in which any non–empty set of (left
or right) ideals that is partially ordered by inclusion, has a minimal
element. In a (left or right) artinian ring any non–invertible element is a
(left or right) zero divisor. As finite rings are always left and right
artinian we get that the neighbor relation is completely determined by the
incidence relation in planes PG($2,R$) over finite rings. In [23] a proof is
given of the property that in a finite ring any left zero divisor is also a
right zero divisor.

######  Theorem 3.1.

(Veldkamp [84]) Let $R$ be a finite ring and PG($2,R$) the projective plane
over $R$. Then two distinct points are neighboring if and only if they are
incident with either no or at least two lines. Two distinct lines are
neighboring if and only if they are incident with either no or at least two
points.

A projective ring plane is called linearly connected (Veldkamp [84]), neighbor
cohesive (Drake and Jungnickel [27]) or punctally cohesive (Baker et al. [9])
if any two distinct points are incident with at least one line. For planes
over rings of stable rank 2 it is proved in Veldkamp that two points are
incident with at least one line if and only if $R$ has the following property:
for any two $r_{1},r_{2}\in R$ there exists $a\in R$ such that
$Rr_{1}+Rr_{2}=R(r_{2}+ar_{1})$. This is fulfilled for $R$ a left Bézout ring,
i.e. a ring for which any finitely generated left ideal is a principal ideal.
Dually two lines are incident with at least one point iff $R$ is a right
Bézout ring. For finite rings the Bézout conditions amount to the condition
that $R$ is a principal ideal ring (recall that for a finite ring the notions
left principal, right principal and principal coincide). So we can reformulate
the theorem for finite rings as follows:

######  Theorem 3.2.

(Veldkamp [84]) Let $R$ be a finite ring and PG($2,R$) the projective plane
over $R$. Then any two points are incident with at least one line (the plane
is linearly connected) and dually if and only if $R$ is a principal ideal
ring.

The possibility of more than one line incident with two neighboring points
(and dually) corresponds to the presence of zero divisors in the ring. So any
two distinct points are incident with exactly one line (and dually) if and
only if $R$ is a Bézout domain. In the finite case this becomes: if $R$ is a
principal ideal domain, hence if $R$ is a finite field. Hence:

######  Theorem 3.3.

(Veldkamp [84]) Let $R$ be a finite ring and PG($2,R$) the projective plane
over $R$. Then any two distinct points are incident with exactly one line and
dually if and only if $R$ is a finite field (i.e. PG($2,R$) is a pappian
projective plane).

Next we look at the special case of (finite) local rings. For such rings the
definition of the projective plane PG($2,R$) and his neighbor relations, can
be adapted (in an equivalent way) a little. E.g. two points $(x,y,z)$ and
$(x^{\prime},y^{\prime},z^{\prime})$ are neighbors if and only if
$(x^{\prime},y^{\prime},z^{\prime})$ $-$ $(x,y,z)\lambda\in J\times J\times J$
for some $\lambda\in R\setminus J$ with $J$ the maximal ideal of $R$ and
similarly for lines.

######  Theorem 3.4.

(Veldkamp [84]) Let $R$ be a (finite) ring and PG($2,R$) the projective plane
over $R$. Then the neighbor relation $\approx$ between points (between lines
resp.) is transitive if and only if $R$ is a (finite) local ring.

For local rings $R$ there is a canonical epimorphism $\varphi$ from $R$ onto
the division ring $\mathbb{K}=R/J$. This epimorphism induces an epimorphism
$\pi$ of the projective plane PG($2,R$) onto the (ordinary) projective plane
PG($2,\mathbb{K}$) by putting $\pi(x,y,z)=(\phi(x),\phi(y),\phi(z))$ and
$\pi[u,v,w]=[\phi(u),\phi(v),\phi(w)]$ and the neighbor relation can be
expressed by means of $\pi$ : $p\sim L$ if and only if $\pi(p)$ I $\pi(L)$ and
similarly $p\sim q$ iff $\pi(p)=\pi(q)$ and $L\sim M$ iff $\pi(L)=\pi(M)$.

The projective plane over a local ring, also known as a (desarguesian)
projective Klingenberg plane, therefore is strongly connected with an ordinary
desarguesian projective plane. One could say that the points (and lines) of an
ordinary projective plane are blown up to clusters of neighboring points
(lines) to produce a projective Klingenberg plane. In the finite case the
epimorphic image of PG($2,R$) is the plane PG($2,q$) over the Galois field
GF($q$).

Combining theorems 3.1, 3.2 and 3.4 yields the following:

######  Theorem 3.5.

(Veldkamp [84]) Let $R$ be a finite ring and PG($2,R$) the projective plane
over $R$. Then the neighbor relation $\approx$ between points (between lines
resp.) is transitive and two neighboring points are incident with at least two
lines and dually, if and only if $R$ is a finite local principal ideal ring.

From section 2 we know that finite local principal ideal rings are synonym for
finite chain rings or finite $H$–rings. Recall that projective planes over
$H$–rings are called (desarguesian) projective Hjelmslev planes.

We now summarize the possibilities for projective planes over a finite ring.

######  Corollary 3.6.

Let $R$ be a finite ring and PG($2,R$) the projective plane over $R$. Then
only four cases are possible:
(a) $R$ has no zero divisors and hence is a field and PG($2,R$) is an ordinary
pappian projective plane (two distinct points are incident with exactly one
line and dually)
(b) $R$ is a local principal ideal ring (hence a chain ring = $H$–ring) and
PG($2,R$) is a desarguesian projective Hjelmslev plane
(c) $R$ is local but not a principal ideal ring and PG($2,R$) is a
desarguesian projective Klingenberg plane (but not a Hjelmslev plane)
(d) $R$ is semilocal (but not local) and PG($2,R$) has non–transitive neighbor
relations

The fourth class (d) is the wildest as it contains all finite rings which are
not local (but necessarily semilocal due to the finiteness, i.e. with a finite
number of maximal ideals). Important examples of rings belonging to this class
are the full matrix rings $M_{n}(q)$ over GF($q$). Projective planes over full
matrix rings were first mentioned by Ree [73] and further studied by J.A. Thas
[80] who also gave an interpretation of PG($2,M_{n}(q)$) in terms of the
projective space PG($3n-1,q$). Other examples are the rings $\mathbb{Z}_{m}$
with $m\not=p^{r}$ (see [29]). Of special interest are also the rings of
double numbers $\mathbb{B}(q)=$GF($q$) + GF($q$)$\,t$ with $t^{2}=t$. They
possess exactly two maximal ideals. In [75] projective planes over
$\mathbb{B}(q)$ are studied.

Examples of finite local rings that are not chain rings (c) are provided by
the rings GF($q$)[$x,y$]/$\langle x^{n},xy,y^{n}\rangle$ ($n>1$). The
corresponding planes are finite desarguesian Klingenberg planes that are not
Hjelmslev planes.

Class (b) contains many interesting examples, including the Galois rings
GR($q^{n},p^{n}$) ($q=p^{r}$) and the rings $\mathbb{A}(p^{r},n)=$
GF($p^{r}$)$[x]/x^{n}$ (called quasi–Galois rings in [14]). The rings
$\mathbb{A}(p^{r},n)$ can also be interpreted as matrix rings, consisting of
all matrices $(a_{ij})$ with elements belonging to GF($q$) and
$a_{i,j+i-1}=a_{1j}$ and $a_{ij}=0$ for $i>j$. For $n=2$ the ring of dual
numbers $\mathbb{D}(q)$ over GF($q$) is included. Projective planes over dual
numbers were considered yet a century ago by Corrado Segre [78].

Class (a) finally consists of all the Galois fields GF($q$) with the
associated projective planes PG($2,q$).

For finite (not necessarily desarguesian) projective Klingenberg and Hjelmslev
planes a unique set of parameters (the order) can be given (see [48] and
[28]): for any flag $(p,L)$ there are exactly $t$ points on $L$ neighboring
with $p$ and exactly $s$ points on $L$ not neighboring with $p$. Moreover: the
number of points = the number of lines = $s^{2}+st+t^{2}$, any line is
incident with $s+t$ points, any point is incident with $s+t$ lines, any point
has $t^{2}$ neighbors, any line has $t^{2}$ neighbors, $t|s$ and
$r=\frac{s}{t}$ is the order of the projective plane that is the canonical
epimorphic image of the Klingenberg plane, and $s\leq t^{2}$ or $t=1$.

For a finite desarguesian projective Klingenberg plane this yields

######  Theorem 3.7.

(Drake and Jungnickel [27] ) Let $R$ be a finite local ring. Then the
projective Klingenberg plane (Hjelmslev plane in some cases) PG($2,R$) has
parameters $s=|R|=q^{n}$ and $t=|J|=q^{n-1}$ with $q=p^{r}$ a prime power.

To conclude this section we consider rings of order 4. This is the smallest
order for which there exist rings that are not division rings. There are four
non–isomorphic rings (with unit) of order 4. The first is the Galois ring
GF($4$) that gives rise to the projective plane PG($2,4$). The second is the
chain ring $\mathbb{Z}_{4}\cong GR(4,4)\cong W_{2}(2)$ with characteristic 4,
the coordinate ring of a projective Hjelmslev plane. The third is the chain
ring $\mathbb{D}(2)\cong\mathbb{A}(2,2)$ of characteristic 2 and with GF($2$)
as a subfield (dual numbers over GF($2$)) that gives rise to another
projective Hjelmslev plane (it is proved in [47] that the plane over
$\mathbb{D}(2)$ is embeddable in PG($5,q$) while the plane over
$\mathbb{Z}_{4}$ is not). Finally the fourth is the non–local ring
$\mathbb{B}(2)\cong GF(2)[t]/(t^{2}-t)$ of characteristic 2 (double numbers
over GF($2$)), associated to a Veldkamp plane with non–transitive neighbor
relation.

##  4 Finite ring–like structures and ring–like planes

Besides finite rings, even more general finite ring–like algebraic structures
deserve a closer look in relationship to geometry. Till now very few research
has been done in this area (except for finite field–like algebraic
structures).
In the literature several generalizations of rings can be found. Among the
most important are: non–associative rings, nearrings and semirings. For all
these structures there are finite examples and a generalization of the concept
“local ring” exists, which opens perspectives for Klingenberg–like geometries
associated to these generalized rings.

###  4.1 Semirings

A semiring is a structure $(S,+,\cdot)$ with $(S,+)$ a commutative semigroup
with identity element $0$, $(S,\cdot)$ a (not necessarily commutative)
semigroup with identity element $1$ ($\not=0$), in which the left and right
distributivity of multiplication over addition hold and in which $0$ is
absorbing for multiplication $a\cdot 0=0\cdot a=0$. Hence semirings differ
from rings by the fact that elements do not always have an inverse for the
addition (the additive group of a ring is replaced by a semigroup).
Semirings were first introduced by Vandiver [83] in 1935 and in the past years
there was an enormous amount of publications on the subject (see e.g. the work
of Glazec [34] for a survey), mainly in relation to computer science and
automata theory but they also are interesting algebraic objects on their own,
see [38] and [35].

Examples of finite semirings are $B(n,i)$ on the set $\\{0,1,\ldots,n-1\\}$
($n\geq 2$ and $0\leq i\leq n-1$) with addition $\oplus$ defined by $a\oplus
b=a+b$ if $0\leq a+b<n$ and $a\oplus b=c$ if $a+b\geq n$ with $c$ the unique
number such that $c\equiv a+b$ (mod $n-i$) and $0\leq c\leq n-1$.
Multiplication $\odot$ is defined in a similar way. In particular $B(n,0)$ is
the ring $\mathbb{Z}_{n}$ of integers modulo $n$ and $B(2,1)$ is known as the
boolean semiring $\mathbb{B}$. For other values of $n$ and $i$ one obtains
semirings that are not rings.

In semirings zero divisors and zero sums are of interest ($a$ is a zero sum of
$S$ if there exists an element $b\not=0$ in $S$ such that $a+b=0$). In [39] it
is proved that if $S$ is a finite commutative semiring then either every zero
sum is a zero divisor or $S$ is a ring. As a corollary one has that a finite
commutative semiring without zero divisors (a semidomain) either is zero sum
free ($a+b=0$ always implies $a=b=0$) or is a finite domain (and hence a
field).

A semiring is called a semifield if $(S^{\ast},\cdot)$, with
$S^{\ast}=S\setminus\\{0\\}$, is a group (in a semifield the multiplication
need not to be commutative, so the term semi division ring would be better).
We have to warn for confusion between the semifields considered here and the
semifields defined in the context of non–desarguesian projective planes by
e.g. Albert, Dickson, Knuth and others and which are also known under the name
division algebras or distributive quasifields. Those are generalizations of
division rings by dropping the need for associativity of the multiplication,
so $(S,+,\cdot)$ is in that context a semifield if $(S,+)$ is a group (which
turns out to be commutative) and $(S^{\ast},\cdot)$ is a loop and the two
distributivity laws of multiplication over addition hold (for a survey on
those semifields, see e.g. [57])

To my knowledge no research has been done yet on geometries over (finite)
semirings that are not rings. So there may be opportunities. In particular a
generalization to Klingenberg–like planes (over local semirings) seems
possible though not trivial as the concept of ideals in semirings is much more
complicated. As a starting point for ideals in semirings and the concept of
local semiring see e.g. [6] and [37].

###  4.2 Nearrings

A left nearring is a structure $(N,+,\cdot)$ with $(N,+)$ a (not necessarily)
commutative group with identity element $0$, $(N,\cdot)$ a (not necessarily
commutative) semigroup with identity element $1$ ($\not=0$), in which the left
distributivity of multiplication over addition holds: $a\cdot(b+c)=a\cdot
b+a\cdot c$. Similar for right nearring. Nearrings differ from rings by the
fact that addition isn’t necessarily commutative and there is only
distributivity on one side. Nearrings which are distributive on both sides are
rings (the commutativity of the addition then follows automatically). Most of
the material on nearrings can be found in the work of Pilz [70] and [71].

A (left or right) nearring is called a (left or right) nearfield if
$(S^{\ast},\cdot)$ is a group, with $S^{\ast}=S\setminus\\{0\\}$. So
nearfields are division rings with only distributivity on one side. Nearfields
were first discovered by Dickson in 1905 and are useful in constructing
examples of non–desarguesian projective planes. All finite nearfields are
classified by Zassenhaus. They are either Dickson–nearfields or they belong to
one of seven exceptional classes.

Little research has been done on geometry over nearrings that are not
nearfields (except for the class of planar nearrings which give rise to
balanced incomplete block designs, but these nearrings don’t possess a
multiplicative identity element and therefore are less usable in the context
of projective geometry). So another suggestion for research could be a
treatment of plane projective geometries over (finite) nearrings that are not
rings. The special case of Klingenberg planes (over local nearrings) and
Hjelmslev planes (over $H$–nearrings) was already initiated in two general
papers by E. Kolb, see [53] and [54] and by Törner in [82]. Local nearrings
were introduced by Maxson in [65] and partially classified in [66]. Among the
results is the fact that the additive group of a finite local nearring always
is a $p$–group and the existence of a natural epimorphism from a local
nearring onto a local nearfield. Other results on finite nearrings can be
found in [2], [20], [21], [22], [45], [58], [59], [64] and [87].

###  4.3 Alternative rings

Finally, we mention some results on non–associative rings. A non–associative
ring is a structure $(A,+,\cdot)$ which satisfies all axioms for an
(associative) ring with multiplicative identity element, except for the
associativity of the multiplication. An alternative ring is a non–associative
ring such that $a\cdot(a\cdot b)=a^{2}\cdot b$ and $(a\cdot b)\cdot b=a\cdot
b^{2}$ for all $a,b\in A$. Alternativity is a weaker condition than
associativity. If any element in an alternative ring $A$ is a unit, then $A$
is called an alternative division ring. Alternative division rings are used to
construct a class of non–desarguesian projective planes, called Moufang
planes. By the theorem of Artin–Zorn every finite alternative division ring is
a field, hence finite Moufang projective planes are desarguesian (and
pappian).

Generalizations to alternative rings that are not division rings are due to
Baker, Lorimer and Lane. In [10] Moufang projective Klingenberg planes are
defined as projective Klingenberg planes that are $(p,L)$–transitive for all
flags and it is proved that they can be coordinatized by a local alternative
ring. In [8] several characterizations of local alternative rings are given
and an analogue for non–associative chain rings and $H$–rings is defined
properly. In the finite case it is proved that the concepts of alternative
$H$–ring, left (or right) alternative chain ring and local alternative
principal ideal ring are equivalent. Moreover the theorem of Artin–Zorn is
expanded : any finite alternative chain ring (or $H$–ring) is associative [9].
Leaving out the condition of being local, leads to more general alternative
rings. It is hard to define projective ring planes over such rings in a
suitable way. Faulkner has done it for alternative stable rank 2 rings in [30]
(generalizing Veldkamp’s results for associative stable rank 2 rings) and for
alternative rings in which any one–sided unit is two–sided in [31]
(generalizing the planes of Barbilian).

## References

  * [1] Y. Al–Khamees, The enumeration of finite principal completely primary rings, Abh. Math. Sem. Univ. Hamburg 51 (1981), 226–231.
  * [2] B. Amberg, P. Hubert and Y. Sysak, Local nearrings with dihedral multiplicative group, Journal of Algebra 273 (2004), 700–717.
  * [3] L. M. Arkhipov, Finite principal ideal rings, Math. Notes, 12 (1972), 656–659.
  * [4] B. Artmann, Desarguessche Hjelmslev-Ebenen $n$-ter Stufe, Mitt. Math. Sem. Giessen, 91 (1971), 1–19.
  * [5] K. Asano, Über Hauptidealringe mit kettensatz, Osaka Math. J. 1 (1949), 52–61.
  * [6] R.E. Atani and S.E. Atani, Ideal theory in commutative semirings, Bull. Acad. Rep. Moldova 2 (57) (2008), 14–23.
  * [7] P. Y. Bacon, Desarguesian Klingenberg planes, Trans. Am. Math. Soc., 241 (1978), 343–355.
  * [8] C.A. Baker, N.D. Lane and J.W. Lorimer, Local alternative rings and finite alternative right chain rings, C. R. Math. Rep. Acad. Sci. Canada, 12 (1990), 53–58.
  * [9] , The Artin–Zorn theorem for finite punctally cohesive projective Klingenberg planes, Ars Combin., 29 (1990), 143–149.
  * [10] , A coordinatization for Moufang Klingenberg planes, Bull. Belg. Math. Soc. – Simon Stevin, 65 (1991), 3–22.
  * [11] D. Barbilian, Zur Axiomatik der projektiven ebenen Ringgeometrien I and II, Jahresbericht Deutsch. Math. Verein 50 (1940), 179–229 and 51 (1941), 34–76.
  * [12] M. Behboodi and R. Beyranvand, On the structure of commutative rings with $p_{1}^{k_{1}}p_{2}^{k_{2}}\cdots p_{n}^{k_{n}}$ $(1\leq k\leq 7)$ zero divisors, European Journal of Pure and Applied Math. 3 (2) (2010), 303–316.
  * [13] F. Bingen, Géométrie projective sur un anneau semi–primaire, Acad. Roy. Belg. Bull. Cl. Sci. 52 (1966), 13–24.
  * [14] G. Bini and F. Flamini, Finite commutative rings and their applications, Kluwer, Boston–Dordrecht–London (2002).
  * [15] H.H. Brungs and G. Törner, Embedding right chain rings in chain rings, Can. J. Math., 30 (5) (1978), 1079–1086.
  * [16] C. J. Chikunji, On a class of finite rings, Comm. Alg., 27 (10) (1999), 5049–5081.
  * [17] C. J. Chikunji, Enumeration of finite rings with Jacobson radical of cube zero arXiv:math/9905030 (1999), 1–20.
  * [18] W.E. Clarke and D.A. Drake, Finite chain rings, Abh. Math. Sem. Univ. Hamburg 39 (1973), 147–153.
  * [19] W.E. Clark and J.J. Liang, Enumeration of finite commutative chain rings, J. Algebra 27 (3) (1973), 445–453.
  * [20] J. Clay, The near–rings of groups of low order, Math. Z. 104 (1968), 364–371.
  * [21] J. Clay and J. Malone, The near–rings with identities on certain finite groups, Math. Scand. 19 (1966), 146–150.
  * [22] J. Clay and D. Doi, The near–rings with identity on alternating groups, Math. Scand. 23 (1968), 54–56.
  * [23] B. Corbas, Rings with a few zero divisors, Math. Ann. 181 (1969), 1–7.
  * [24] B. Corbas and P.D. Williams, Rings of order $p^{5}$, Part I and II, J. Algebra 231 (2000), 677–690 and 691–704.
  * [25] A. Cronheim, Dual numbers, Witt vectors, and Hjelmslev planes, Geom. Ded. 7 (1978), 287–302.
  * [26] P. De Winne, 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 (3-4) (1974), 139–159.
  * [27] D.A. Drake and D. Jungnickel, Finite Hjelmslev planes and Klingenberg epimorphisms, in Rings and Geometry, NATO Adv. Study Inst., Istanbul, eds R. Kaya et al., Reidel, Dordrecht, (1985), 153–231.
  * [28] D.A. Drake and H. Lenz, Finite Klingenberg planes, Abh. Math. Sem. Univ. Hamburg, 44 (1975), 70–83.
  * [29] F. Eugeni and E. Galiè, Sui piani costruiti su anelli, Dipartimento M.E.T., Università di Teramo, Italy, (1991), 143–162.
  * [30] J.R. Faulkner, Coordinatization of Moufang–Veldkamp planes, Geom. Ded. 14 (1983), 189–201.
  * [31] , Barbilian planes, Geom. Ded. 30 (1989), 125–181.
  * [32] J.L. Fisher, Finite principal ideal rings, Canad. Math. Bull., 19 (3) (1976), 277–283.
  * [33] G. Ganske and B. R. McDonald, Finite local rings, Rocky Mountain J. Math, 3 (4) (1973), 512–540.
  * [34] K. Glazec, A short guide through the literature on semirings, Math. Inst. Univ. Wroclaw, Poland (1985).
  * [35] J. Golan, Semirings and their applications, Kluwer, Dordrecht (1999).
  * [36] M. González, On distinguishing local finite rings from finite rings only by counting elements and zero divisors, European Journal of Pure and Applied Math. 7 (1) (2014), 109–113.
  * [37] V. Gupta en R. Chaudhari, Right local semirings, Asian-European Journal of Mathematics 6 (1) (2013), 1–5.
  * [38] U. Hebisch and H. J. Weinert, Semirings and semifields, Handbook of Algebra part I, Elsevier, Amsterdam (1996), 425–462.
  * [39] A. Hetzel and R. Lewis Lufy, On the relationship between zero–sums and zero–divisors of semirings, Kyungpook J. Math. 49 (2009), 221–233.
  * [40] J.W.P. Hirschfeld, Projective geometries over finite fields, Oxford University Press, New–York (1998).
  * [41] J. Hjelmslev, Einleitung in die allgemeine Kongruenzlehre. III, Danske Vid. Selsk. Mat.-Fys. Medd. 19 (1942), nr. 12.
  * [42] T. Honold and I. Landjev, Linear codes over finite chain rings, Electr. J. Comb. 7 (2000), # R11.
  * [43] T. Honold and I. Landjev, Codes over rings and ring geometries, in Current research topics in Galois geometry, eds. L. Storme and J. de Beule, Nova Science Publishers, New–York 7 (2011), 159–184.
  * [44] X.-D. Hou, Finite commutative chain rings, Finite Fields and Their Applications 7 (2001), 382–396.
  * [45] P. Hubert, Nearrings and a construction of triply factorized groups, Ph.D. Thesis, Universität Mainz (2005).
  * [46] G. Janusz, Separable algebras over commutative rings, Trans. Am. Math. Soc., 122 (1966), 461–479.
  * [47] D. Keppens and H. Van Maldeghem Embeddings of projective Klingenberg planes in the projective space PG($5,\mathbb{K}$), Contrib. Alg. Geom., 50 (2) (2009), 483–493.
  * [48] E. Kleinfeld, Finite Hjelmslev planes, Illinois J. Math., 3 (1959), 403–407.
  * [49] W. Klingenberg, Projektive und affine Ebenen mit Nachbarelementen, Math. Z. 60 (1954), 384–406.
  * [50] , Euklidische Ebenen mit Nachbarelementen, Math. Z. 61 (1954), 1–25.
  * [51] , Desarguessche Ebenen mit Nachbarelementen, Abh. Math. Sem. Univ. Hamburg 20 (1955), 97–111.
  * [52] , Projektive Geometrien mit Homomorphismus, Math. Ann 132 (1956), 180–200.
  * [53] E. Kolb, Projective Klingenberg planes over nearrings, Journal of Geometry 46 (1993), 82–91.
  * [54] , Hjelmslev planes over nearrings, Discrete Mathematics 155 (1996), 147–155.
  * [55] F. Knüppel, Projective planes over rings, Resultate Math., 12 (1987), 348–356.
  * [56] W. Krull, Algebraische theorie der Ringe I, II and III, Math. Annalen 8 (1923), 80–122, 91 (1924), 1–46 and 92 (1924), 183–213.
  * [57] M. Lavrauw and O. Polverino, Finite semifields and Galois geometry, in Current research topics in Galois Geometry, ed. De Beule and Storme, NOVA Academic Publishers (2011).
  * [58] S. Ligh, Near–rings with descending chain condition, Compositio Mathematica 21 (2) (1969), 162–166.
  * [59] S. Ligh and J. J. Malone, Zero divisors and finite nearrings, 6 (1) (2013), 1–5.
  * [60] J.W. Lorimer, Structure theorems for commutative Hjelmslev rings with nilpotent radicals, C. R. Math. Rep. Acad. Sci. Canada, 6 (3) (1984), 123–127.
  * [61] , Affine Hjelmslev rings and planes, Annals of Discr. Math., 37 (1988), 265–276.
  * [62] , The classification of compact right chain rings, Forum Math., 4 (1992), 335–347.
  * [63] J.W. Lorimer and N.D. Lane, Desarguesian affine Hjelmslev planes, J. für die reine und angew. Math., 1 (1975), 336–352.
  * [64] C. Maxson, On finite nearrings with identity, Amer. Math. Monthly 74 (1967), 1228–1230.
  * [65] , On local near–rings, Math. Z. 106 (1968), 197–205.
  * [66] , Local near–rings of cardinality $p^{2}$, Canad. Math. Bull. 11 (4) (1968), 555–561.
  * [67] B. R. McDonald, Finite rings with identity, Marcel Dekker, New York (1974).
  * [68] A.A. Nechaev, Finite rings with applications, in Handbook of Algebra, vol. 5, Elsevier (2008), 213–320.
  * [69] A. Neumaier, Nichtkommutative Hjelmslev-Ringe, Festband für H. Lenz, Freie Universität Berlin, (1976), 200–213.
  * [70] G. Pilz, Near–rings, North–Holland, Amsterdam, 2nd Edition (1983).
  * [71] , Nearrings and nearfields, Handbook of Algebra part I, Elsevier, Amsterdam (1996), 463–498.
  * [72] R. Raghavendran, Finite associative rings, Compositio Mathematica 21 (1969), 195–229.
  * [73] R. Ree, On projective geometry over full matrix rings, Trans. Am. Math. Soc. 6 (1) (1955), 144–150.
  * [74] A.S. Rybkin, Finite local rings of principal ideals, Math. Notes 28 (1981), 465–472.
  * [75] M. Saniga and M. Planat, Projective Planes Over Galois double numbers and a geometrical principle of complementarity, arXiv:math/0601261v3, (2006), 1–9.
  * [76] M. Satyanarayana Characterization of local rings, Tôhoku Math. Journal, 19 (1967), 411–416.
  * [77] B. Segre, Le geometrie di Galois, Ann. Math. Pura e Appl. 48 (1959), 1–96.
  * [78] C. Segre, Le geometrie proiettive nei campi di numeri duali, Atti Accad. Sci. Torino, 47 (1911), 114–133 and 164–185.
  * [79] L.A. Skornyakov, Rings chain–like from the left (Russian), Izv. Vyssh. Uchebn. Zaved. Mat., 4 (1966), 114–117.
  * [80] J. A. Thas, 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 (1971), 459–532.
  * [81] G. Törner, Eine klassifizierung von Hjelmslev–ringen und Hjelmslev–Ebenen, Mitt. Math. Sem. Giessen, 107 (1974), 1–77.
  * [82] , Über den Stufenaufbau von Hjelmslev–Ebenen, Mitt. Math. Sem. Giessen, 126 (1977), 1–43.
  * [83] H. S. Vandiver, Note on a simple type of algebra in which cancellation law of addition does not hold, Bull. Amer. Math. Soc. 40 (1934), 914–920.
  * [84] F.D. Veldkamp, Projective planes over rings of stable rank 2, Geom. Dedicata 11 (1981), 285–308.
  * [85] , Projective ring planes: some special cases, in Atti Conv. Geometria combinatoria e di incidenza, La Mendola, 1982, Rend. Sem. Mat. Brescia 7 (1984), 609–615.
  * [86] , Geometry over rings, Handbook of Incidence Geometry, Elsevier, Amsterdam (1995), 1033–1084.
  * [87] G. Wendt, On zero divisors in near–rings, Int. Journal of Algebra 3 (2009), 21–32.
  * [88] R.S. Wilson, On the structure of finite rings I, Comp. Math 26 (1973), 79–93.
  * [89] , On the structure of finite rings II, Pacific J. Math. 51 (1974), 317–325.
  * [90] B. R. Wirt, Finite non–commutative local rings, Ph.D. Thesis, University of Oklahoma (1972).
  * [91] T. Wu, H. Yu and D. Lu, The structure of finite local principal ideal rings, arXiv:math/11055179v3, (2011).