On Centrally Essential Rings

Oleg Lyubimtsev111Nizhny Novgorod State University, Nizhny Novgorod, Russia;
email<EMAIL_ADDRESS>., Askar Tuganbaev222National Research
University MPEI, Moscow, Russia; Lomonosov Moscow State University, Moscow,
Russia<EMAIL_ADDRESS>.

Abstract. A ring $R$ with center $C$ is said to be centrally essential if the
module $R_{C}$ is an essential extension of the module $C_{C}$. In this paper,
we study properties of ideals of centrally essential rings, centrally
essential quaternion algebras, and group rings of Hamiltonian groups.

Key words. centrally essential ring, quaternion algebra, group ring

The work of Oleg Lyubimtsev is supported by Ministry of Education and Science
of the Russian Federation, project FSWR-2023-0034. The study of Askar
Tuganbaev is supported by grant of Russian Science Foundation (=RSF), project
22-11-00052, https://rscf.ru/en/project/22-11-00052.

MSC2020 database 16D25, 16R99

##  1 Introduction

We consider only associative rings which are not necessarily unital. A ring
$R$ is said to be centrally essential if $R$ is either commutative or for any
its non-central element $a$, there exist non-zero central elements $x$ and $y$
with $ax=y$. It is clear that a unital ring $R$ with center $C(R)$ is
centrally essential if and only if the module $R_{C(R)}$ is an essential
extension of the module $C(R)_{C(R)}$. An unital ring $R$ is centrally
essential if and only if for any non-zero element $a\in R$, there exist non-
zero central elements $x,y\in R$ with $ax=y$. The last condition does not hold
for any non-zero ring with zero multiplication (it is centrally essential). We
note that any commutative ring is centrally essential.

In Section 2, we study properties of ideals of centrally essential rings which
are semiprime as rings. We prove that ideals (resp., right ideals) of a
centrally essential ring, which are semiprime as rings, are contained in the
center of the ring (resp., are reduced commutative ideals); see Theorem 2.2
and Proposition 2.4. As a corollary, we obtain that every minimal non-
nilpotent right ideal of a centrally essential ring is contained in the center
of the ring.

In Section 3, we describe centrally essential quaternion algebras and
centrally essential group rings of Hamiltonian groups (Theorems 3.2 and 3.6).

In Section 4, we prove commutativity of a ring which is a rational extension
of its center (Proposition 4.2) and give examples of non-commutative centrally
essential rings with injective right maximal ring of fractions (Example 4.3)
and a non-commutative centrally essential semiring of order 5 (Example 4.4).

We denote by $Q_{cl}(R)$ and $Q_{max}(R)$ the classical ring of fractions and
the maximal (right) ring of fractions of the ring $R$, correspondingly. A
right ideal $I$ of the ring $R$ is said to be essential if $I\cap J\neq 0$ for
all non-zero right ideals $J$ of $R$. We denote by $\text{Ann}_{R}(S)$ the
left annihilator $\\{r\in R\,|\,rS=0\\}$ of the set $S$ in the ring $R$. The
right annihilator of the set $S$ is similarly defined. The right singular
ideal $Z_{r}(R)$ is the set of all elements whose right annihilators are
essential. A ring $R$ is said to be right nonsingular if $Z_{r}(R)=0$. A ring
is called a (right) ring of fractions of $R$ if it contains $R$ and is
contained in $Q_{max}(R)$. It is well known that a ring $R$ has a von Neumann
regular maximal ring of fractions if and only if $R$ is right nonsingular;
e.g., see [6, Proposition 8.3]. A ring is said to be right (resp., left)
strongly bounded if every right (resp., left) ideal contains a non-zero ideal;
e.g., see [2]. A ring is said to be strongly bounded if is right and left
bounded. We set $[a,b]=ab-ba$ for any two elements $a,b$ of the ring $R$.

A ring $A$ is said to be right invariant (resp., right quasi-invariant) if
every right ideal (resp., maximal right ideal) of $A$ is an ideal of $A$.

We denote by $Q_{8}$ the quaternion group of order 8, i.e., the group with two
generators $a$, $b$ and defining relations $a^{4}=1$, $a^{2}=b^{2}$ and
$aba^{-1}=b^{-1}$; e.g., see [7, Section 4.4]. We have:

$Q_{8}=\\{e,a,a^{2},b,ab,a^{3},a^{2}b,a^{3}b\\},$

the center $Z(Q_{8})=\\{e,a^{2}\\}$.

##  2 Properties of Right Ideals which Are Semiprime as Rings

We recall that a module $X$ is called an essential extension of a submodule
$Y$ in $X$ if $Y\cap Z\neq 0$ for any non-zero submodule $Z$ in $X$. In this
case, $Y$ is called an essential submodule of the module $X$. A right ideal
$I$ of the ring $R$ is said to be reduced if $I$ does nor contain non-zero
nilpotent elements.

Lemma 2.1. Let $R$ be a centrally essential ring and let $J$ be a right ideal
in $R$ which is semiprime as a ring. Then $J$ is a reduced right ideal.

Proof. Let $0\neq a\in J$ and $a^{2}=0$. Since $R$ is a centrally essential
ring, $0\neq ac=d\in C(R)\cap J$, where $c\in C(R)$. Then $dR$ is an ideal in
$J$ and $(dR)^{2}=0$. This is a contradiction. $\square$

Theorem 2.2. Let $R$ be a centrally essential ring and let $I$ be an ideal in
$R$ which is semiprime as a ring. Then $I\subseteq C(R)$. In particular, $I$
is a centrally essential ring.

Proof. We prove that $I$ is a commutative ideal. Let $[a,b]\neq 0$ for some
$a,b\in I$. It follows from [1, Lemma 1.1(iv)] that $C(R)\cap I=C(I)$. Then
$J=\\{c\in C(R)\,|\,ac\in C(I)\\}$ is a non-zero ideal in $C(R)$. Let $d\in
C(I)$ be with $dJ=0$. If $ad\neq 0$, then $(ad)z\in C(I)\backslash\\{0\\}$ for
some $z\in C(R)$. Then $dz\in J$ and $d(dz)=(dz)^{2}=0$. It follows from Lemma
2.1 that $dz=0$, whence $adz=0$. This is a contradiction. Therefore, $ad=0$
and $d\in J$. Then $d^{2}=0$ and $d=0$. Consequently,
$\text{Ann}_{C(I)}(J)=0$, where $\text{Ann}_{C(I)}(J)=\\{d\in
C(I)\,|\,dJ=0\\}$. Further, we note that for any $j\in J$, we have $aj=ja\in
C(R)$. Then

$[a,b]j=(ab-ba)j=a(bj)-b(aj)=ajb-ajb=0.$

Therefore, $[a,b]J=0$. In addition, $c_{1}[a,b]=c_{2}\in
C(I)\backslash\\{0\\}$ for some $c_{1}\in C(R)$. Then $c_{2}J=c_{1}[a,b]J=0$
and $\text{Ann}_{C(I)}(J)\neq 0$. This is a contradiction. Consequently,
$[a,b]=0$ and $I$ is a commutative ideal. Since $I=C(I)=I\cap C(R)$, we have
$I\subseteq C(R)$. $\square$

Corollary 2.3. Every minimal non-nilpotent right ideal $J$ of a centrally
essential ring $R$ is contained in the center of the ring $R$ and is ideal in
$R$.

Proof. In [12, Theorem 1.7(1)], it is proved that $J$ is a two-sided ideal.
Since $J^{2}\neq 0$, the ideal $J$ is semiprime as ring. $\square$

Proposition 2.4. Let $R$ be a centrally essential ring and let $J$ be a right
ideal in $R$ which is semiprime as a ring. Then $J$ is a reduced commutative
right ideal.

Proof. It follows from Lemma 2.1 that $J$ is a reduced right ideal. In [12,
Proposition 2.6.], it is proved that any centrally essential ring is strongly
bounded. It follows from [2, Proposition 1] that $J$ is an essential extension
of some ideal $I$. By Theorem 2.2, the ideal $I$ is contained in the center
rings $R$. We note that $\text{Ann}_{J}(I)=\\{j\in J\,|\,jI=0\\}=0$, otherwise
$\text{Ann}_{J}(I)\cap I\neq 0$, and $I$ is not a reduced ring. Consequently,
for every $j\in J$, there exists an element $i\in I$ such that $0\neq ji\in
I$. Then $J$ is a centrally essential semiprime ring which have to be
commutative; see [11, Proposition 2.1]. $\square$

Under an additional condition, the right ideal $J$ from Proposition 2.4 is
contained in the center of the ring.

Corollary 2.5. Let $R$ be a centrally essential ring and let $J$ be a right
ideal of $R$ which is semiprime, as a ring. If $\text{Ann}_{R}(J)=0$, then
$J\subseteq C(R)$.

Proof. Let $x\in R$ and $a,b\in J$. It follows from Proposition 2.4 that

$b(ax-xa)=b(ax)-b(xa)=(ba)x-(bx)a=a(bx)-a(bx)=0.$

Therefore, $ax-xa=t\in\text{Ann}_{R}(J)$, where $\text{Ann}_{R}(J)$ is the
right annihilator $J$. In addition, $\text{Ann}_{R}(J)$ also is the left
annihilator. Indeed, if $tJ\neq 0$, then $(tJ)^{2}=0$ which contradicts to the
property that $J$ is reduced. Therefore, $t=0$ and $J\subseteq C(R)$.
$\square$

Proposition 2.6. If $R$ is a centrally essential ring and $I$ is a minimal
non-nilpotent right ideal, then $I$ and $R/I$ are centrally essential
rings333We recall that $I$ is an ideal in $R$, by Corollary 2.3..

Proof. By assumption, $I=eR$ for some idempotent $e\in C(R)$; see [13, Lemma
2.3]. Then $R=eR\oplus(1-e)R$ is a ring direct sum. Consequently, $R$ is a
centrally essential ring if and only if $I=eR$ and $J=(1-e)R$ are centrally
essential rings. $\square$

We note that Proposition 2.6 is not true for a nilpotent minimal ideal. Let
$Q_{8}$ be the quaternion group, $\widehat{Q}_{8}=\sum_{g\in Q_{8}}g$,
$R=\mathbb{Z}_{2}Q_{8}$, and let $A$ be the factor ring of the centrally
essential group ring $R$ with respect to the least non-zero ideal
$I=\\{0,\widehat{Q}_{8}\\}$. In the paper [12, Example 3.2], it is proved that
the ring $A$ is not centrally essential.

Corollary 2.7. Let $R$ be a ring, in which all idempotents are central, and
let $I$ be a minimal non-nilpotent right ideal in $R$. A ring $R$ is centrally
essential if and only if $I$ and $R/I$ are centrally essential rings.

Classes of rings in which idempotents are central include, for example,
invariant rings or $UC$-rings; see [19, Proposition 1] and [16, Lemma 4],
correspondingly. From [19, Proposition 2], we obtain the following corollary.

Corollary 2.8. Any non-nilpotent minimal ideal of an invariant centrally
essential ring is a field.

We note that there exist centrally essential rings which are not left or right
invariant (quasi-invariant); see [10, Example 2.4] and [12, Theorem 1.7(2)].

Open Question 2.9. Is it true that a nilpotent minimal ideal of a centrally
essential ring is contained in the center of the ring?

##  3 Centrally Essential Quaternion Algebras and Group Rings

The following assertion is well known, but it is given with a proof for
convenience.

Lemma 3.1. Let $R$ be a ring and let
$\\{M_{\alpha}\\}_{\alpha\in\mathcal{A}}$,
$\\{N_{\alpha}\\}_{\alpha\in\mathcal{A}}$ be two sets of right $R$-modules
such that for any $\alpha\in\mathcal{A}$, the module $N_{\alpha}$ is an
essential submodule of the module $M_{\alpha}$. Then the module
$N=\oplus_{\alpha\in\mathcal{A}}N_{\alpha}$ is an essential submodule of the
module $M=\oplus_{\alpha\in\mathcal{A}}M_{\alpha}$.

Proof. For any $\alpha\in\mathcal{A}$, let $\pi_{\alpha}$ be the canonical
epimorphism from $M$ onto $M_{\alpha}$. For any $x\in M$, we set

$k(x)=|\\{\alpha\in\mathcal{A}:\pi_{\alpha}(x)\not\in N_{\alpha}\\}|.$

The following properties of the function $k(x)$ are obvious:
$\forall x\in M,\quad k(x)<\infty$;
$\forall x\in M,\;t\in R,\quad k(tx)\leq k(x)$;
$\forall x\in M,\quad k(x)=0\Leftrightarrow x\in N$.

Now we prove that $xR\cap N\neq 0$ provided $x\in M\setminus\\{0\\}$. We
assume the contrary. Among elements $x\in M\setminus\\{0\\}$ with $xR\cap
N=0$, we take an element $x_{0}$ with least value of the parameter $k(x)$. It
is clear that $k(x_{0})>0$, since $x_{0}\not\in N$. Therefore, there exists a
subscript $\overline{\alpha}\in\mathcal{A}$ such that
$\pi_{\overline{\alpha}}(x_{0})\not\in N_{\overline{\alpha}}$. Since
$N_{\overline{\alpha}}$ is essential in $M_{\overline{\alpha}}$, there exists
an element $a_{0}\in R$ such that $\pi_{\overline{\alpha}}(x_{0})a_{0}\in
N_{\overline{\alpha}}\setminus\\{0\\}$. Then it is clear that $x_{0}a_{0}\neq
0$ and $k(a_{0}x_{0})<k(x_{0})$. Therefore, it follows from the choice of the
element $x_{0}$ that $x_{0}a_{0}R\cap N\neq 0$. Since $a_{0}R\subseteq R$, we
have $x_{0}R\cap N\neq 0$, this is a contradiction. Consequently, $N$ is an
essential submodule in $M$. $\square$

Let $A$ be a commutative ring, $I=Ann_{A}(2)$, and let $a,b$ be two invertible
elements of the ring $A$. By following [20], we denote by $R=(a,b,A)$ the
algebra over $A$ with basis $\\{1,i,j,k\\}$ such that the multiplication in
$R$ is defined by relations

$i^{2}=a,\;j^{2}=b,\;ij=-ji=k,\;ik=-ki=aj,\;kj=-jk=bi.$

It is known that the center of the ring $R$ is of the form $C(R)=A\cdot
1+I\cdot i+I\cdot j+I\cdot k$; see [20, Lemma 2(b)].

Theorem 3.2. The following conditions are equivalent:

$(a)$ the ring $(a,b,A)$ centrally essential;

$(b)$ $Ann_{A}(2)$ is an essential ideal of the ring $A$.

Proof. $(b)\Rightarrow(a)$. We set $\mathcal{A}=\\{0,1,2,3\\}$,
$M_{0}=N_{0}=A\cdot 1$, $M_{1}=A\cdot i$, $M_{2}=A\cdot j$, $M_{3}=A\cdot k$,
$N_{1}=I\cdot i$, $N_{2}=I\cdot j$, and $N_{3}=I\cdot k$. We have
$C(R)=N_{0}\oplus N_{1}\oplus N_{2}\oplus N_{3}$, and it follows from the
condition $(b)$ that $C(R)$ is an essential $A$-submodule of the module $R$;
$C(R)$ especially is an essential $C(R)$-submodule of $R$.

$(a)\Rightarrow(b)$. Let $t\in A\setminus\\{0\\}$. We consider the element
$r=2t\cdot i\in R$. If $2t=0$, then $t\in At\cap I$, whence $At\cap I\neq 0$.
Otherwise, there exists an element $c\in C(R)$ such that $0\neq cr\in C(R)$.
We have $c=c_{0}\cdot 1+c_{1}\cdot i+c_{2}\cdot j+c_{3}\cdot k$, where
$c_{0}\in A$, $c_{1},c_{2},c_{3}\in I$. Consequently, $cr=2c_{0}t\cdot i$,
whence $2c_{0}t\neq 0$. Then $2c_{0}t\in I$, whence $0\neq 2c_{0}t\in{}At\cap
I$. We have that $At\cap I\neq 0$ in any case, i.e., condition $(b)$ holds.
$\square$

In the paper [18, Theorem 4], it is proved that if $F$ is a field of
characteristic 0 and $G$ is any group, then the group ring $FG$ is right
nonsingular. In the following proposition, we extend this result to
commutative domains of characteristic $0$.

Proposition 3.3. Let $R$ be a commutative domain of characteristic $0$. Then
the group ring $RG$ is right nonsingular for any group $G$.

Proof. Let $Q=Q_{cl}(R)$ be the field of fractions of the ring $R$. We note
that $Q=Q_{max}(R)$; e.g., see [17, Exercise 2, P. 167]. It follows from [3,
3.3] that $QG\subseteq Q_{max}(RG)$. In addition, the group ring $QG$ is
nonsingular; see [18, Theorem 4]. Since $QG$ is the ring of fractions of $RG$,
we have that $QG$ and $RG$ have the same maximal ring of fractions which is
von Neumann regular. Consequently, the group ring $RG$ is nonsingular.
$\square$

Centrally essential group rings over fields were studied in the paper [13],
where it was proved that centrally essential group algebras over fields of
characteristic $0$ are commutative. From [14, Proposition 2.8], we obtain the
following corollary.

Corollary 3.4. Let $R$ be a commutative domain of characteristic $0$ and let
$G$ be any group. The group ring $RG$ is centrally essential if and only if
the ring $RG$ is commutative. $\square$

We also have a similar assertion for non-commutative domains of characteristic
$0$.

Proposition 3.5. Let $R$ be a domain of characteristic $0$ and let $G$ be any
group. The group ring $RG$ is centrally essential if and only if $RG$ is
commutative.

Proof. If $charR=0$ and $R$ is a domain, then $RG$ does not have nil-ideals;
in particular, it is semiprime, see [21, Theorem 5.1]. It follows from [13,
Proposition 3.3] that $RG$ is a commutative ring. $\square$

We recall that a group $G$ is said to be Hamiltonian if $G$ is not Abelian and
all subgroups of $G$ are normal. If $H$ is a subgroup of the group $G$, then
we denote by $\Delta(G,H)$ the fundamental ideal of the subgroup $H$ in the
group ring $RG$.

We recall that $Q_{8}$ denotes the quaternion group of order 8.

Theorem 3.6. Let $G$ be a Hamiltonian group and let $R$ be a unital ring. The
group ring $RG$ is centrally essential if and only if $R$ is a centrally
essential ring of characteristic $2$.

Proof. It is well known444See [7, 12.5.4]. that a Hamiltonian group is the
direct product $G=Q_{8}\times A\times B$, where $A$ is an elementary Abelian
$2$-group and $B$ is an Abelian group in which every element is of finite
order. Then

$RG=R(Q_{8}\times A\times B)\cong R(A\times B)Q_{8}.$

It follows from [13, Lemma 2.1 and Lemma 2.2] that the group ring $R(A\times
B)Q_{8}$ is centrally essential if and only if $RQ_{8}$ is a centrally
essential group ring. Therefore, the study of central essentiality of the ring
$RG$ is reduced to the case $G=Q_{8}$.

Let $RQ_{8}$ be a centrally essential ring. Then $R$ is a centrally essential
ring; see [13, Lemma 2.1]. We assume that $charR>2$. Then $RQ_{8}\cong
R(Q_{8}/Q^{\prime}_{8})\oplus\Delta(Q_{8},Q^{\prime}_{8})$ is a direct sum;
see [15, Proposition 3.6.7]. Therefore, $RQ_{8}$ is a centrally essential ring
if and only if $R(Q_{8}/Q^{\prime}_{8})$ and $\Delta(Q_{8},Q^{\prime}_{8})$
are centrally essential rings. Since $Q_{8}/Q^{\prime}_{8}$ is an Abelian
group, $R(Q_{8}/Q^{\prime}_{8})$ is a centrally essential ring; see [13, Lemma
2.2]. We consider the ring $\Delta(Q_{8},Q^{\prime}_{8})$ with basis
$\\{f,af,bf,abf\\}$, where $f=1-e_{Q^{\prime}_{8}}=\frac{1}{2}(1-a^{2})$. We
describe the center $C=C(\Delta(Q_{8},Q^{\prime}_{8}))$ of the ring
$\Delta(Q_{8},Q^{\prime}_{8})$. Let
$\alpha=\alpha_{0}f+\alpha_{1}af+\alpha_{2}bf+\alpha_{3}abf\in C$,
$\beta=\beta_{0}f+\beta_{1}af+\beta_{2}bf+\beta_{3}abf\in\Delta(Q_{8},Q^{\prime}_{8})$.
The condition $\alpha\beta=\beta\alpha$ holds if and only if relations

$\left\\{\begin{aligned} \alpha_{2}\beta_{3}-\alpha_{3}\beta_{2}=0\\\
\alpha_{1}\beta_{3}-\alpha_{3}\beta_{1}=0\\\
\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}=0\\\ \end{aligned}\right.$

hold. If $\alpha_{i}\neq 0$ for $i\in\\{1,2,3\\}$, then by setting
$\beta_{i}=0$, we obtain $\alpha_{i}\beta_{j}=0$, where $j\neq i$. Then
$\alpha_{i}=0$ for $\beta_{j}=1$; this is a contradiction. Consequently,
$C(\Delta(Q_{8},Q^{\prime}_{8}))=\\{\alpha_{0}f\,\mid\,\alpha_{0}\in C(R)\\}$.
Therefore, for the element $af\in\Delta(Q_{8},Q^{\prime}_{8})\setminus C$,
there does not exist an element $c\in C$ such that $0\neq af\cdot c\in C$.
Therefore, $\Delta(Q_{8},Q^{\prime}_{8})$ is not a centrally essential ring.
Therefore, $RQ_{8}$ is not a centrally essential ring. This is a
contradiction.

Conversely, let $R$ be a centrally essential ring of characteristic $2$. It
follows from [4, Theorem 9] that the fundamental ideal
$\Delta=\Delta(Q_{8},Q_{8})$ is nilpotent. Next we use the proof method from
the paper [13, Proposition 2.6]. For $0\neq x\in RQ_{8}$, we consider all
possible products $x(1-z)$, where $z\in Z(Q_{8})$. If at least one of them,
for example $x_{1}=x(1-z_{1})$, is not equal to zero, then we consider
$x_{1}(1-z)$, and so on. This process will terminate after a finite number of
steps, i.e., there exists a positive integer $k$ such that
$x_{k}\cdot\Delta(Q_{8},Z(Q_{8}))=0$ (we assume that that $x_{0}=x$). Then
$x_{k}\in RQ_{8}\sum_{Z(Q_{8})}$; e.g., see [21, Proposition 1.12]. Therefore,
$RQ_{8}\sum_{Z(Q_{8})}\subseteq C(RQ_{8})$. Indeed, if $g,h\in Q_{8}$, then

$[g,h\sum_{Z(Q_{8})}]=[g,h]\sum_{Z(Q_{8})}=gh(1-h^{-1}g^{-1}hg)\sum_{Z(Q_{8})}=0,$

since $Q_{8}$ is a nilpotent group of nilpotence index $2$ and
$h^{-1}g^{-1}hg\in Q^{\prime}_{8}=Z(Q_{8})$. By setting
$c=(1-z_{1})\ldots(1-z_{k})$ (or $c=1$, for $k=0$), we obtain $c\in C(RQ_{8})$
and $0\neq xc=x_{k}\in C(RQ_{8})$. Consequently, $RQ_{8}$ is a centrally
essential ring. $\square$

Example 3.7. Let $R=\mathbb{Z}_{9}$. We consider the following matrix
representation of the ideal $\Delta(Q_{8},Q^{\prime}_{8})$:

$M_{\Delta}=\begin{pmatrix}q_{0}&q_{1}&q_{2}&q_{3}\\\
-q_{1}&q_{0}&q_{3}&-q_{2}\\\ -q_{2}&-q_{3}&q_{0}&q_{1}\\\
-q_{3}&q_{2}&-q_{1}&q_{0}\end{pmatrix}.$

As proven above, $M_{\Delta}$ is not a centrally essential ring. Moreover,
$M_{\Delta}$ has non-central idempotents, for example

$e=\begin{pmatrix}5&1&1&0\\\ -1&5&0&-1\\\ -1&0&5&1\\\ 0&1&-1&5\end{pmatrix}.$

It follows from [13, Lemma 2.3] that $\mathbb{Z}_{9}Q_{8}$ is not a centrally
essential ring. $\square$

##  4 Supplements and Remarks

A right module $M$ over a ring $A$ is a rational extension of its submodule
$N$ if for every submodule $B$ such that $N\subseteq B\subseteq M$ and for any
$f\in Hom_{R}(B,M)$, the condition $f(N)=0$ holds if and only if $f=0$. This
is equivalent to the property that for any $x,0\neq y\in M$, there exist $r\in
R$ and $n\in\mathbb{Z}$ such that $xr+xn\in N$ and $yr+yn\neq 0$, e.g., see
[6, Proposition 7.2].

A ring $A$ is said to be centrally rational if the module $A_{C}$ is a
rational extension of the module $C_{C}$.

Remark 4.1. We note that all centrally rational rings are centrally essential.
The converse is not true. Indeed, if $F$ is a field $\mathbb{Z}/2\mathbb{Z}$
and $A$ is a group $F$-algebra of the group quaternion of order 8, then $A$ is
a non-commutative centrally essential ring, while every centrally rational
ring is commutative, by the following proposition.

Proposition 4.2. A ring $A$ is commutative if and only if $A$ is a centrally
rational ring.

Proof. Let $ab\neq ba$ for some $a,b\in R$. Then $ac\in C$ and $(ab-ba)c\neq
0$ for some $c\in C$. Then $abc=(ac)b=bac$. This is a contradiction. $\square$

It is well known that if a ring $R$ is nonsingular, then the ring $Q_{max}(R)$
is injective; e.g., see [6, Theorem 1+2, P. 69]. However, if $Z_{r}(R)\neq 0$,
then $Q_{max}(R)$ could also be injective.

Example 4.3. We give an example of a non-commutative centrally essential ring
with right injective maximal ring of fractions.

We consider the group algebra $R=\mathbb{Z}_{2}Q_{8}$ with center

$C(R)=\\{\lambda_{0}+\lambda_{1}a^{2}+\lambda_{2}(a+a^{3})+\lambda_{3}(b+a^{3}b)+\lambda_{4}(ab+a^{3}b),\quad\lambda_{i}\in\mathbb{Z}_{2}\\};$

e.g., see [17, Part 2]. Then $|C(R)|=32$, $R$ is a ring with subdirectly
indecomposable center and core $H=\\{0,\widehat{Q}_{8}\\}$, where
$\widehat{Q}_{8}=\sum_{g\in Q_{8}}g$. In addition, $R$ is a centrally
essential ring, by Theorem 3.6. It follows from [9, Corollary 13.5.5(5),
Theorem 13.5.7] that $R$ is a $QF$-ring. Then the module $R_{R}$ is injective;
see [9, Theorem 13.2.1]. It follows from [6, Theorem B, P. 64] that
$Q_{max}(R)=R$. It follows from 4.2 that the ring $R$ is not centrally
rational, which can be verified directly. Indeed, let $0\neq a\in H$, $b\in
Q_{8}$. We note that $bc\notin C(R)$ for all invertible central elements $c$
(it is easy to see that there are 16 such elements, and they are sums of an
odd number of conjugacy classes as terms). Since $R$ is a centrally essential
ring, $bd\in C(R)$ for some non-invertible central element $d$. On the other
side, $d$ is a nilpotent element and $dH=0$; see [5, Theorem 2.1(v)].
Consequently, $R$ is not a centrally rational ring. $\square$

The group algebra $\mathbb{Z}_{2}Q_{8}$ contains 256 elements. In the paper
[11, Example 4.1, $\mathbb{F}=\mathbb{Z}_{2}$], it is constructed a non-
commutative centrally essential ring consisting of 128 elements. At the
present time, the authors do not know examples of non-commutative centrally
essential rings, containing less than 128 elements. A different situation
occurs when considering centrally essential semirings. We recall that a
semiring is a structure that differs from an associative ring, because the
additive operation is not allways reversible. The zero of a semiring $S$ is
multiplicative by definition: for any $s\in S$, we have $0s=s0=0$. A semiring
$S$ is said to be semisubtractive if for any distinct $a,b\in S$, there exists
an element $x\in S$ such that $a+x=b$ or $b+x=a$, e.g., see [8].

Example 4.4. We consider a semiring $S=\\{0,1,a,b,c\\}$ in which operations
are defined as follows:

$\begin{tabular}[]{ |c|c|c|c|c|c|}\hline\cr$+$&$0$&$1$&$a$&$b$&$c$\\\
\hline\cr$0$&$0$&$1$&$a$&$b$&$c$\\\ \hline\cr$1$&$1$&$1$&$1$&$b$&$1$\\\
\hline\cr$a$&$a$&$1$&$a$&$b$&$a$\\\ \hline\cr$b$&$b$&$b$&$b$&$b$&$b$\\\
\hline\cr$c$&$c$&$1$&$a$&$b$&$c$\\\
\hline\cr\end{tabular},\quad\quad\begin{tabular}[]{
|c|c|c|c|c|c|}\hline\cr$\cdot$&$0$&$1$&$a$&$b$&$c$\\\
\hline\cr$0$&$0$&$0$&$0$&$0$&$0$\\\ \hline\cr$1$&$0$&$1$&$a$&$b$&$c$\\\
\hline\cr$a$&$0$&$a$&$a$&$a$&$c$\\\ \hline\cr$b$&$0$&$b$&$b$&$b$&$c$\\\
\hline\cr$c$&$0$&$c$&$c$&$c$&$c$\\\ \hline\cr\end{tabular}.$

The center $C(S)$ of the semiring $S$ consists of elements $0$, $1$ and $c$.
Since $0\neq x\cdot c=c\in C(S)$ for $0\neq x\in S$, we have that $S$ is a
non-commutative semisubtractive centrally essential semiring of order 5.

## References

  * [1] Armendariz E. P., Birkenmeier G. F., Park J. K. Ideal intrinsic extensions with connections to $PI$-rings // J. Pure Appl. Algebra. – 2009. – Vol. 213. – P. 1756–1776.
  * [2] Birkenmeier G.F., Tucci R.P. Homomorphic images and the singular ideal of a strongly right bounded ring // Comm. Algebra. – 1988. – Vol. 16. – no. 6. – P. 1099–1122.
  * [3] Burgess W. D. Rings of quotients of group rings // Canad. J. Math. – 1969. – Vol. 21, no. 4. – P. 865–875.
  * [4] Connell I. G. On the group ring // Canad. J. Math. – 1963. – Vol. 15, no. 4. – P. 650–685.
  * [5] Deshpande M.G. Structure of right subdirectly irreducible ring I // J. Algebra. – 1971. – Vol. 17. – P. 317–325.
  * [6] Faith C., Lectures on injective modules and quotient rings, Lecture Notes in Mathematics, Vol. 49, Berlin/Heidelberg/New York, Springer, 1967.
  * [7] Hall M., The Theory of Groups, Macmillan, New York, 1959.
  * [8] Hebisch U., Weinert H. J. Semirings and semifields // in M. Hazewinkel (ed .): Handbook of Algebra, North-Holland, Amsterdam. – 1996. – Vol. 1. – P. 425-462.
  * [9] Kasch F., Modules and rings, London Mathematical Society monograph, no. 17, 1982.
  * [10] Lyubimtsev O.V., Tuganbaev A.A. Centrally essential torsion-free rings of finite rank // Beitr. Algebra Geom.– 2021. – Vol. 62. – no. 3. – P. 615–622.
  * [11] Lyubimtsev O.V., Tuganbaev A.A. Local Centrally Essential Subalgebras of Triangular Algebras // Linear and Multilinear Algebra. – 2022. – Vol. 70. – no 13. – P. 2415–2424.
  * [12] Lyubimtsev O.V., Tuganbaev A.A. Centrally essential factor rings and subdirect indecomposability // Comm. Algebra. – published on-line, https://doi.org/10.1080/00927872.2023.2267688.
  * [13] Markov V.T., Tuganbaev A.A. Centrally essential group algebras // J. Algebra. – 2018. – Vol. 512. – P. 109–118.
  * [14] Markov V.T., Tuganbaev A.A. Rings essential over their centers // Comm. Algebra. – 2019. – Vol. 47, no. 4. – P. 1642–1649.
  * [15] Milies C.P., Sehgal S.K. An Introduction to Group Rings// Kluwer Academic Publishers, Dordrecht, 2002.
  * [16] W.K. Nicholson W.K., Zhou Y. Rings in which elements are uniquely the sum of an idempotent and a unit // Glasgow Math. J. – 2004. – Vol. 46. – P. 227–236.
  * [17] Passman D.S. The Algebraic Structure of Group Rings// John Wiley and Sons, New York, 1977.
  * [18] Snider R.L. On the singular ideal of a group algebra // Comm. Algebra. – 1976. – Vol. 4, no. 11. – P. 1087–1089.
  * [19] Thierrin G. On duo rings // Can. Math. Bull. – 1960. – Vol. 3 – P. 167–172.
  * [20] Tuganbaev A.A. Quaternion algebras over commutative rings // Mathematical Notes. – 1993. – Vol. 53, no. 2. – P. 204–207.
  * [21] Zalesskii A.E., Mikhalev A.V. Group rings // Journal of Soviet Mathematics. – 1975. – Vol. 4. – P. 1–78.


# A New Structure in the Deuteron

Misak M. Sargsian and Frank Vera  Florida International University, Miami,
Florida 33199,USA

###### Abstract

We demonstrate that a paradigm shift from considering the deuteron as a system
of bound proton and neutron to considering it as a pseudo-vector system in
which we observe proton and neutron, results in a possibility of probing a new
“incomplete” P-state like structure on the light-front (LF), at extremely
large internal momenta, which can be achieved in high energy transfer electro-
disintegration of the deuteron. Investigating the deuteron on the light-front,
where the vacuum-fluctuations are suppressed, we found that this new
structure, together with conventional S- and D- states, is a leading order in
transferred energy of the reaction, thus it is not suppressed on the light-
front. The incompleteness of the observed P-state results in a violation of
angular condition which can happen only if deuteron contains non-nucleonic
structures such as $\Delta$$\Delta$, $N^{*}N$ or hidden color components. We
demonstrate that experimentally verifiable signatures of “incomplete” P-states
are angular anisotropy of LF momentum distribution of the nucleon in the
deuteron as well as an enhancement of the tensor polarization strength beyond
the S- and D- wave predictions at large internal momenta in the deuteron.

One of the outstanding issues of strong interaction physics is the
understanding of the dynamics of transition between hadronic to quark-gluon
phases of matter. Such transitions at high temperature is relevant to the
evolution of the universe after the big bang and can be studied experimentally
in heavy ion collisions. Transitions at low (near zero) temperatures and high
densities (“cold-dense” transitions) are relevant for superdense nuclear
matter that can exists at the cores of neutron stars and can set up the limits
of matter density before it collapses to the black hole. However the direct
exploration of “cold-dense” transitions is severely restricted.

Currently the accepted ways of investigating such transitions are;
(1) Studying nuclear medium modification of quark-gluon structure of bound
nucleons: Such a modification was discovered in 1983 - by European Muon
CollaborationEuropeanMuon:1983wih  \- commonly referred to as EMC effect. Few
progresses were made in understanding of this phenomena for past 40 years (for
reviews see Frankfurt:1988nt ; Hen:2016kwk ), including the observation of the
dependence of the effect on local nuclear densitySeely:2009gt  and the
important role of short range nucleonic correlations in the EMC effect for
medium to large nucleiWeinstein:2010rt ; CLAS:2019vsb . In all these cases the
role of the hadronic to quark-gluon transition is not clearly understood.
(2) Studying the implications of the transition of baryonic matter to the
quark matter in the cores of neutron stars. The situation with the existence
of quark matter in the cores of Neutron Stars even more unclear than with the
EMC effect. With the observation of unexpectedly large neutron star
massesFonseca:2021wxt  ($\approx 2.08M_{\odot}$) it was expected that if such
stars would have radii, $R<10$ km it will be indicative of large quark matter
component in their cores. However observed radiuses for the large mass neutron
stars are above $R\geq 12$ km (e.g. Ref.Miller:2021qha ).

While progress in advancing the studies of EMC effects is seen in performing
new generation of experiments in which density of nuclear medium is controlled
by tagging a recoil nucleon which is in short range correlation with the
probed nucleon (e.g. Ref.Melnitchouk:1996vp ). The neutron star studies rely
on improving of detection techniques that will allow to identify anomalously
small size neutron stars.

In the present work we are suggesting a new method of studying baryon-quark
transition using simplest known atomic nucleus, the deuteron.

Deuteron on the Light Front (LF): Our current mindset about deuteron is fully
non-relativistic, within which, the observation that it has total spin, $J=1$
and positive parity, $P$, together with the relation that for non-relativistic
wave function, $P=(-1)^{l}$, one concludes that the deuteron consists of S-
and D- partial waves for proton-neutron system.

However if we are interested in deuteron structure at internal momenta
comparable with the nucleon rest mass then nonrelativistic framework is not
valid and the problem is more fundamental, related to the description of a
relativistic bound system. This situation is similar to the description of
quark structure of nucleon in QCD in which case due to the small masses of u-
and d- quarks the vacuum fluctuations may overshadow the composite structure
of the nucleon (see e.g. Ref.Feynman:1973xc ).

To discuss relativistic structure of the deuteron on needs to identify the
process in which the deuteron structure is probed. In our case we consider
high-momentum transfer electrodisintegration process:

$e+d\rightarrow e^{\prime}+p+n\vspace{-0.2cm}$ (1)

in which one of the nucleons are struck by the incoming probe and the
spectator nucleon is probed with momenta comparable to nucleon masses. If one
can neglect (or remove) the effects related to final state interactions of two
outgoing nucleons, then the above reaction at high $Q^{2}$, measures the
probability of observing proton and neutron in the deuteron at very large
relative momenta. In such a formulation the deuteron is not a composite system
consisting of proton and neutron but it is a composite pseudo - vector ($J=1$,
$P=+$) “particle” from which one extracts proton and neutron. How such a
proton and neutron are produced at such extremal conditions is related to the
dynamical structure of LF deuteron wave function, which may include internal
elastic $pn\rightarrow pn$ as well as inelastic $\Delta\Delta\to pn$,
$N^{*}N\to pn$ or $N_{c}N_{c}\to pn$ transitions. Here, $\Delta$ and $N^{*}$
denote $\Delta$-isobar and $N^{*}$ resonances, while $N_{c}$ is a color octet
baryonic state contributing to the hidden-color component in the deuteron.

The framework for calculation of reaction (1) in relativistic domain is LF
approach (e.g. Ref.Frankfurt:1981mk ; Miller:2000kv ; Brodsky:1997de ;
Frederico:1991vb ; Vera:2021rnw ; inprogress ) in which one introduces LF
deuteron wave function in the form:

$\psi_{d}^{\lambda_{d}}(\alpha_{i},p_{\perp},\lambda_{1}\lambda_{2})=-{\bar{u}(p_{2},\lambda_{2})\bar{u}(p_{1},\lambda_{1})\Gamma^{\mu}_{d}\chi_{\mu}^{\lambda_{d}}\over{1\over
2}(m_{d}^{2}-4{m_{N}^{2}+p_{\perp}^{2}\over\alpha_{i}(2-\alpha_{i})})\sqrt{2(2\pi)^{3}}},\vspace{-0.3cm}$
(2)

where $\alpha_{i}=2{p_{i+}\over p_{d+}}$, ($i=1,2$) and
$\alpha_{1}+\alpha_{2}=2$ are LF momentum fractions of two nucleons outgoing
from deuteron that has four-momentum $p_{d}^{\mu}$.
Absorbing the energy denominator into the vertex function and using crossing
symmetry one obtains:

$\displaystyle\psi_{d}^{\mu}(\alpha_{i},p_{\perp},\lambda_{1},\lambda_{2})=-\bar{u}(p_{2},\lambda_{2})\Gamma^{\mu}_{d}(k){(i\gamma_{2}\gamma_{0})\over\sqrt{2}}\bar{u}(p_{1},\lambda_{1})^{T}$
$\displaystyle=-\sum\limits_{\lambda_{1}^{\prime}}\bar{u}(p_{1},\lambda_{1})\Gamma^{\mu}_{d}\gamma_{5}{\epsilon_{\lambda_{1},\lambda_{1}^{\prime}}\over\sqrt{2}}u(p_{1},\lambda^{\prime}_{1}),\vspace{-2.cm}$
(3)

where $u(p,\lambda)$’s are LF bi-spinors of proton and neutronLepage:1980fj
and $\epsilon_{i,j}$ is two dimensional Levi-Civita tensor, with $i,j=\pm 1$
helicity of nucleon. Since the deuteron is a pseudo-vector “particle”, due to
$\gamma_{5}$ in Eq.(3), the vertex $\Gamma^{\mu}_{d}$ is a four-vector which
we can construct in a general form that explicitly satisfies time reversal,
parity and charge conjugate symmetries. Noticing that at the $d\to pn$ vertex
on the light-front the ”-” ($p_{-}=E-p_{z}$) components of the four-momenta of
the particles are not conserved, in addition to the four-momenta of two
nucleons, $p_{1}^{\mu}$ and $p_{2}^{\nu}$ one has additional four-momentum:

$\Delta^{\mu}\equiv
p_{1}^{\mu}+p_{2}^{\mu}-p_{d}^{\mu}\equiv(\Delta^{-},\Delta^{+},\Delta_{\perp})=(\Delta^{-},0,0),$
(4)

where

$\displaystyle\Delta^{-}=p_{1}^{-}+p_{2}^{-}-p_{d}^{-}={m_{N}^{2}+k^{2}_{\perp}\over
p_{1}^{+}}+{m_{N}^{2}+k^{2}_{\perp}\over p_{2}^{+}}-{M_{d}^{2}\over
p_{d}^{+}}$ $\displaystyle={1\over
p_{d}^{+}}\left[{4(m_{N}^{2}+k^{2}_{\perp})\over\alpha_{1}(2-\alpha_{1})}-M_{d}^{2}\right]={4\over
p_{d}^{+}}\left[m_{N}^{2}-{M_{d}^{2}\over 4}+k^{2}\right].\vspace{-0.6cm}$ (5)

Here $k$ is the relative momentum in the $pn$ CM system defined as:

$k=\sqrt{{m_{N}^{2}+k_{\perp}^{2}\over\alpha_{1}(2-\alpha_{1})}-m_{N}^{2}}\ \
\ \mbox{and}\ \ \ \alpha_{1}={E_{k}+k_{z}\over E_{k}},$ (6)

where $E_{k}={m^{2}+k^{2}}$. With $p_{1}^{\mu}$, $p_{2}^{\mu}$ and
$\Delta^{\mu}$ 4-vectors the $\Gamma_{d}^{\mu}$ 4-vector function is
constructed in the following form:

$\displaystyle\Gamma_{d}^{\mu}=\Gamma_{1}\gamma^{\mu}+\Gamma_{2}\frac{(p_{1}-p_{2})^{\mu}}{2m_{N}}+\Gamma_{3}\frac{\Delta^{\mu}}{2m_{N}}+\Gamma_{4}\frac{(p_{1}-p_{2})^{\mu}\Delta\hskip-6.0pt/}{4m_{N}^{2}}$
$\displaystyle+i\Gamma_{5}\frac{1}{4m_{N}^{3}}\gamma_{5}\epsilon^{\mu\nu\rho\gamma}(p_{d})_{\nu}(p_{1}-p_{2})_{\rho}(\Delta)_{\gamma}+\Gamma_{6}\frac{\Delta^{\mu}\Delta\hskip-6.0pt/}{4m_{N}^{2}},$
(7)

where $\Gamma_{i}$, ($i=1,6$) are scalar functions describing dynamics of $pn$
component being observed in the deuteron.

High Energy Approximation: For large $Q^{2}$ limit, LF momenta for reaction
(1) are chosen as follows:

$\displaystyle
p_{d}^{\mu}\equiv(p_{d}^{-},p_{d}^{+},p_{d\perp})=\left({Q^{2}\over
x\sqrt{s}}\left[1+{x\over\tau}-\sqrt{1+{x^{2}\over\tau}}\right]\right.,$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.{Q^{2}\over
x\sqrt{s}}\left[1+{x\over\tau}+\sqrt{1+{x^{2}\over\tau}}\right],0_{\perp}\right)$
$\displaystyle q^{\mu}\equiv(q^{-},q^{+},q_{\perp})=\left({Q^{2}\over
x\sqrt{s}}\left[1-x+\sqrt{1+{x^{2}\over\tau}}\right],\right.$ $\displaystyle\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left.{Q^{2}\over
x\sqrt{s}}\left[1-x-\sqrt{1+{x^{2}\over\tau}}\right],0_{\perp}\right),$ (8)

where $s=(q+p_{d})^{2}$, $\tau={Q^{2}\over M_{d}^{2}}$ and $x={Q^{2}\over
M_{d}q_{0}}$, with $q_{0}$ being virtual photon energy in the deuteron rest
frame. The high energy nature of this process results in,
$p_{d}^{+}\sim\sqrt{Q^{2}}\gg m_{N}$. Then one observes in Eq.(5) that the
$\Delta^{-}$ term is suppressed by the large $p_{d}^{+}$ factor.

Analyzing now the vertex function (7) one observes that ${\Delta^{-}\over
2m_{N}}$ enters as a small parameter in the problem in which $\Gamma_{3}$ and
$\Gamma_{4}$ terms enter with the order of ${\cal O}^{1}({\Delta^{-}\over
2m_{N}})$ while $\Gamma_{6}$ term enters as ${\cal O}^{2}({\Delta^{-}\over
2m_{N}})$. Situation with $\Gamma_{5}$ term, is however different; since for
the covariant components: $\Delta_{+}={1\over 2}\Delta^{-}$, $p_{d,-}={1\over
2}p_{d}^{+}$, the term with $\epsilon^{\mu+\perp-}$ is a leading order (${\cal
O}^{0}({\Delta^{-}\over 2m_{N}})$) due to the fact that large $p_{d}^{+}$
factor is cancelled in $p_{d,-}\Delta_{+}={1\over 4}p_{d}^{+}\Delta^{-}$
combination.

Keeping the leading, ${\cal O}^{0}(\Delta^{-})$, terms in Eq.(7) the LF
deuteron wave function reduces toinprogress ; Vera:2021rnw :

$\displaystyle\psi_{d}^{\lambda_{d}}(\alpha_{i},p_{\perp})=-\sum\limits_{\lambda_{2},\lambda_{1},\lambda_{1}^{\prime}}\bar{u}(p_{2},\lambda_{2})\left\\{\Gamma_{1}\gamma^{\mu}+\Gamma_{2}{(p_{1}-p_{2})^{\mu}\over
2m_{N}}\right.$ $\displaystyle+\left.\sum\limits_{i=1}^{2}i\Gamma_{5}{1\over
8m^{3}_{N}}\epsilon^{\mu+i-}p^{+}_{d}k_{i}\Delta^{-}\right\\}\gamma_{5}{\epsilon_{\lambda_{1},\lambda_{i}^{\prime}}\over\sqrt{2}}u(p_{1},\lambda^{\prime}_{1})\chi_{\mu}^{\lambda_{d}},$
(9)

where $k_{i}={(p_{1,i}-p_{2,i})\over 2}$, for $i=1,2$. The deuteron’s
polarization four-vector is chosen as:

$\chi_{\mu}^{\lambda_{d}}=(\chi_{0}^{\lambda_{d}},\chi_{\perp}^{\lambda_{d}},\chi_{z}^{\lambda_{d}})=({p_{12}s_{d,z}\over
M_{12}},s_{s,\perp},{E_{12}s_{d,z}\over M_{12}}),$ (10)

where ${\bf p_{12}}=(p_{1_{z}}+p_{2,z},0_{\perp})$,
$E_{12}=\sqrt{M_{12}^{2}+p_{12}^{2}}$ and
$M^{2}_{12}=s_{NN}=4{(m_{N}^{2}+k_{\perp}^{2})\over\alpha_{1}(2-\alpha_{1})}$.
Since the wave function in Eq.(9) is Lorentz boost invariant along the $z$
axis, it is convenient to calculate it in the deuteron CM frame obtained by
boosting with velocity $v={{\bf p_{12}}\over E_{12}}$. Such a transformation
results ininprogress :

$\displaystyle\psi_{d}^{\lambda_{d}}(\alpha_{i},k_{\perp})=-\sum\limits_{\lambda_{2},\lambda_{1},\lambda_{1}^{\prime}}\bar{u}(-k,\lambda_{2})\left\\{\Gamma_{1}\gamma^{\mu}+\Gamma_{2}{{\tilde{k}}^{\mu}\over
m_{N}}+\right.$ $\displaystyle\left.\sum\limits_{i=1}^{2}i\Gamma_{5}{1\over
8m^{3}_{N}}\epsilon^{\mu+i-}p^{\prime+}_{d}k_{i}\Delta^{\prime-}\right\\}\gamma_{5}{\epsilon_{\lambda_{1},\lambda_{i}^{\prime}}\over\sqrt{2}}u(k,\lambda^{\prime}_{1})s_{\mu}^{\lambda_{d}},$
(11)

where $\tilde{k}^{\mu}(0,k_{z},k_{\perp})$ with $k_{\perp}=p_{1\perp}$,
$k^{2}=k_{z}^{2}+k^{2}_{\perp}$ and $E_{k}={\sqrt{S_{NN}}\over 2}$ and
$s_{\mu}^{\lambda_{d}}=(0,{\bf s^{\lambda}_{d}})$ in which:

$s_{d}^{1}=-{1\over\sqrt{2}}(1,i,0),\ s_{d}^{1}={1\over\sqrt{2}}(1,-i,0)\
s_{d}^{0}=(0,0,1).$ (12)

In Eq.(11) “primed” variables correspond to the Lorentz boosts of respective
unprimed quantities:

$p^{\prime+}_{d}=\sqrt{s_{NN}},\
\Delta^{\prime-}={1\over\sqrt{s_{NN}}}\left[{4(m_{N}^{2}+k_{\perp}^{2})\over\alpha_{1}(2-\alpha_{1})}-M_{d}^{2}\right].\vspace{-0.1cm}$
(13)

Since the term related to the $\Gamma_{5}$ is proportional to
${4(m_{N}^{2}+k_{\perp}^{2})\over\alpha_{1}(2-\alpha_{1})}-M_{d}^{2}$ which
diminishes at small momenta, only the $\Gamma_{1}$ and $\Gamma_{2}$ terms will
contribute in nonrelativistic limit defining the $S$\- and $D$\- components of
the deuteron. Thus LF wave function in Eq.(11) provides a smooth transition to
the non-relativistic deuteron wave function. This can be seen by expressing
Eq.(11) through two-component spinors:

$\displaystyle\psi_{d}^{\lambda_{d}}(\alpha_{1},k_{t},\lambda_{1},\lambda_{2})=\sum\limits_{\lambda_{1}^{\prime}}\phi^{\dagger}_{\lambda_{2}}\sqrt{E_{k}}\left[{U(k)\over\sqrt{4\pi}}{\bf\sigma
s_{d}^{\lambda_{d}}}\right.-$
$\displaystyle-\left.{W(k)\over\sqrt{4\pi}\sqrt{2}}\left({3{\bf(\sigma
k)(ks_{d}^{\lambda})}\over k^{2}}-{\bf\sigma s_{d}^{\lambda}}\right)+\right.$
$\displaystyle\left.(-1)^{1+\lambda_{d}\over
2}P(k)Y_{1}^{\lambda_{d}}(\theta,\phi)\delta^{1,\mid\lambda_{d}\mid}\right]{\epsilon_{\lambda_{1},\lambda_{1}^{\prime}}\over\sqrt{2}}\phi_{\lambda^{\prime}_{1}}.$
(14)

Here the first two terms have explicit $S$\- and $D$\- structures were radial
functions defined as:

$\displaystyle U(k)=$ $\displaystyle{2\sqrt{4\pi}\sqrt{E_{k}}\over
3}\left[\Gamma_{1}(2+{m_{N}\over E_{k}})+\Gamma_{2}{k^{2}\over
m_{N}E_{k}}\right]$ (15) $\displaystyle W(k)=$
$\displaystyle{2\sqrt{4\pi}\sqrt{2E_{k}}\over 3}\left[\Gamma_{1}(1-{m_{N}\over
E_{k}})-\Gamma_{2}{k^{2}\over m_{N}E_{k}}\right].$

This relation is known for $pn$-component deuteron wave
functionFrankfurt:1981mk ; Carbonell:1995yi , which allows to model LF wave
function through known radial $S$\- and $D$\- wave functions estimated at LF
relative momentum $k$ defined in Eq.(6).

However in addition to $S$-, $D$\- terms, our observation is that due to the
$\Gamma_{5}$ term, there is an additional leading contribution, which because
of the relation $Y^{\pm}_{1}(\theta,\phi)=\mp i\sqrt{3\over
4\pi}\sum\limits_{i=1}^{2}{(k\times s_{d}^{\pm 1})_{z}\over k}$, has a
$P$-wave like structure, where $P$\- radial function is defined as:

$\displaystyle P(k)=$
$\displaystyle\sqrt{4\pi}{\Gamma_{5}(k)\sqrt{E_{k}}\over\sqrt{3}}{k^{3}\over
m_{N}^{3}}.$ (16)

It is worth emphasizing that this term is purely relativistic in origin: as it
follows from Eq.(16) it has an extra ${k^{2}\over m_{N}^{2}}$ factor in
addition to the ${k^{l=1}\over m_{N}}$ term characteristic to the radial
$P$-wave. As a result one has a smooth transition to $S$\- and $D$-states in
nonrelativistic limit.

The interesting feature of the above result which we will discuss in the next
section, is that the $P$-wave is “incomplete”, that it contributes only for
$\lambda_{d}=\pm 1$ polarizations of the deuteron.

Closing this section we would like to mention that consideration of six
invariant vertex functions and contribution of $P$-radial waves in
relativistic description of the deuteron were discussed earlier in the
literature, see e.g. Refs.Buck:1979ff ; Carbonell:1998rj . However, to the
best of our knowledge the observation that $\Gamma_{5}$ is a leading term on
the light-front (while $\Gamma_{3,4,6}$ terms are suppressed) in high energy
limit and it results in a non-complete $P$-wave contribution are original
results of the present work.

Light Front Density Matrix of the Deuteron:Using Eq.(14) one defines
unpolarized deuteron light-front density matrix in the formFrankfurt:1981mk ;
Frankfurt:1988nt :

$\vspace{-0.3cm}\rho_{d}({\alpha,k_{\perp}})={n_{d}(k,k_{\perp})\over
2-\alpha},$ (17)

where LF momentum distribution is expressed through the radial wave functions
as follows:

$\displaystyle\vspace{-0.4cm}n_{d}(k,k_{\perp})$ $\displaystyle=$
$\displaystyle{1\over
3}\sum\limits_{\lambda_{d}=-1}^{1}\mid\psi_{d}^{\lambda_{d}}(\alpha,k_{\perp})\mid^{2}=$
(18) $\displaystyle=$ $\displaystyle{1\over
4\pi}\left(U(k)^{2}+W(k)^{2}+{k_{\perp}^{2}\over k^{2}}P^{2}(k)\right).$

The LF density matrix satisfies baryonic and momentum sum rules as follows:

$\int\rho_{d}({\alpha,k_{\perp}}){d\alpha\over\alpha}=1\ \ \ \mbox{and}\ \ \
\int\alpha\rho_{d}({\alpha,k_{\perp}}){d\alpha\over\alpha}=1.$ (19)

From the above, the normalization condition for the radial wave functions is:

$\int\left(U(k)^{2}+W(k)^{2}+{2\over 3}P^{2}(k)\right)k^{2}dk=1.$ (20)

The $\Gamma_{5}$-Term and Non-Nucleonic Component in the Deuteron:The unusual
result of Eq.(14) is that the $P$-wave like term enters only for deuteron
polarizations, $\lambda_{d}=\pm 1$. The later is the reason that momentum
distribution in Eq.(18) depends explicitly on the transverse component of the
relative momentum on the light front. Such a behavior is impossible for non-
relativistic quantum mechanics of the deuteron since in this case the
potential of the interaction is real (no inelasticities) and the solution of
Lippmann-Schwinger equation for partial S- and D-waves satisfies “angular
condition”, according to which the momentum distribution in unpolarized
deuteron depends on the magnitude of relative momentum only. Our result do not
contradict the properties of non-relativistic deuteron wave function since as
it was discussed earlier according to Eq.(16) the P-wave is purely
relativistic in its nature. On the other hand, in the relativistic domain the
definition of the interaction potential is not straightforward to allow to use
quantum-mechanical arguments in claiming that momentum distribution in Eq.(18)
should satisfy the angular condition (i.e. depends on magnitude of $k$ only).

For the relativistic domain, on the light-front, the analogue of Lippmann-
Schwinger equation is the Weinberg type equationWeinberg:1966jm  using which
for $NN$ scattering amplitude, in which only nucleonic degrees are considered,
in the CM of the NN system, one obtainsFrois:1991wc :

$\displaystyle T_{NN}(\alpha_{i},k_{i\perp},\alpha_{f},k_{f,\perp})\equiv
T_{NN}(k_{i,z},k_{i\perp},k_{f,z},k_{f,\perp})=$ $\displaystyle
V(k_{i,z},k_{i\perp},k_{f,z},k_{f,\perp})+\int
V(k_{i,z},k_{i\perp},k_{m,z},k_{m,\perp})\times$
$\displaystyle\times{d^{3}k_{m}\over(2\pi)^{3}\sqrt{m^{2}+k_{m}^{2}}}{T_{NN}(k_{m,z},k_{m\perp},k_{f,z},k_{f,\perp})\over
4(k_{m}^{2}-k_{f}^{2})},$ (21)

where “i”, “m” and “f”, subscripts correspond to initial, intermediate and
final $NN$ states respectively and momenta $k_{i,m,f}$ are defined similar to
Eq.(6). The realization of the angular condition for relativistic case will
require that light-front potential to satisfy a condition

$V(k_{i,z},k_{i\perp},k_{m,z},k_{m,\perp})=V(\vec{k}_{i}^{2},(\vec{k}_{m}-\vec{k}_{i})^{2}).$
(22)

Such a conditions is obvious for on-shell limit, since the Lorentz invariance
of the $T_{NN}$ amplitude requires:

$T_{NN}^{on\ shell}(k_{i,z},k_{i\perp},k_{m,z},k_{m,\perp})=T^{on\
shell}_{NN}(\vec{k}_{i}^{2},(\vec{k}_{m}-\vec{k}_{i})^{2})$ (23)

and existence of the Born term in Eq.(21) indicates that the potential $V$
satisfies the same condition in the on-shell limit.

For the off-shell potential the angular condition is not obvious. However in
Ref.Frankfurt:1988nt ; FSMS90 ; Frois:1991wc  it was shown that requirements
of potential $V$ satisfying angular condition in the on-shell limit and that
it can be constructed through the series of elastic $pn$ scatterings result in
a potential which is analytic function of angular momentum. Then with the
minimal assumption that the potential, analytically continued to the complex
angular momentum space, does not diverge exponentially, it was shown that $V$
and the $T_{NN}$ functions satisfy angular condition (Eqs.(22,23)) in general.
Then, using the same potential to calculate LF deuteron wave function will
result in a momentum distribution that will depend on the magnitude of the
relative $pn$ momentum only. This observation requires the restriction by the
$pn$ component only in the deuteron.

Inclusion of the inelastic transitions will completely change the LF equation
for the $pn$ scattering. For example, contribution of $N^{*}N$ transition to
the elastic $NN$ scattering:

$\displaystyle T_{NN}(k_{i,z},k_{i\perp},k_{f,z},k_{f,\perp})=\int
V_{NN^{*}}(k_{i,z},k_{i\perp},k_{m,z},k_{m,\perp})$
$\displaystyle\times{d^{3}k_{m}\over(2\pi)^{3}\sqrt{m^{2}+k_{m}^{2}}}{T_{N^{*}N}(k_{m,z},k_{m\perp},k_{f,z},k_{f,\perp})\over
4(k_{m}^{2}-k_{f}^{2}+m^{2}_{N*}-m_{N}^{2})},$ (24)

will not require the condition of Eq.(22) with the transition potential having
also an imaginary component. Eq.(24 can not be described with any combination
of elastic $NN$ interaction potentials that satisfies the angular condition.
The same will be true also for $\Delta\Delta\to NN$ and $N_{c},N_{c}\to NN$
transitions. This indicates that if the $\Gamma_{5}$ term is not zero then it
should originate from non-nucleonic component in the deuteron.

Estimate of the Possible Effects: Our prediction is that the observation of
anisotropic LF momentum distribution depending on the center of mass $k$ and
$k_{\perp}$ separately will indicate the presence of non-nucleonic component
in the deuteron. Since this effect is due to the $P$-wave like structure,
(originating from $\Gamma_{5}$ term) which has an extra ${k^{2}\over
m^{2}_{N}}$ factor (Eq.(15)) compared to $S$\- and $D$\- radial waves, one
expects it to become important at $k>m_{N}$.

To give qualitative estimate of the possible effects we evaluate $\Gamma_{5}$
vertex function assuming two color-octet baryon transition to the $pn$ system
($N_{c}N_{c}\to pn$) through one-gluon exchange, parameterizing it in the
dipole form ${A\over(1+{k^{2}\over 0.71})^{2}}$. The parameter $A$ is
estimated by assuming 1% contribution to the total normalization from the $P$
wave in Eq.(20). The latter is consistent with the experimental estimation in
Ref.deltadelta  of 0.7%. In Fig.1 we consider dependence of the momentum
distribution of Eq.(18) as a function of $\cos{\theta}={(\alpha-1)E_{k}\over
k}$ for different values of $k$. Notice that if momentum distribution is
generated by $pn$ component only, the angular condition is satisfied, and no
dependence should be observed.

Figure 1: LF momentum distribution of the deuteron as a function of
$\cos{\theta}$, for different values of $k$. Dashed lines - deuteron with $pn$
component only, solid lines - with $P$-wave like component included.

As the figure shows on may expect measurable angular dependence at $k\gtrsim
1$ GeV/C which is consistent with the expectation that inelastic transition in
the deuteron corresponding to the non-nucleonic components takes place at
$k\gtrsim 800$ MeV/c. Additionally due to the fact that the $P$-component
contributes only for $\lambda_{d}=\pm 1$ polarization of the deuteron
(Eq.(14)) one expects enhanced effect in the asymmetry from scattering off the
tensor polarized deuteron:

$A_{T}={n_{d}^{\lambda_{d}=1}(k,k_{\perp})+n_{d}^{\lambda_{d}=-1}(k,k_{\perp})-2n_{d}^{\lambda_{d}=0}(k,k_{\perp})\over
n_{d}(k,k_{\perp})}.$ (25)

As Fig.2 shows the presence of non-nucleonic component will be visible already
at $k\approx 800$ MeV/c resulting in a qualitative difference in asymmetry at
larger momenta as compared with the asymmetry predicted by deuteron wave
function with $pn$-component only.

Figure 2: Tensor asymmetry as a function of $\cos{\theta}$ for different $k$.
Dashed lines - deuteron with $pn$ component only, solid lines - with $P$
component included.

Outlook on Experimental Verification of the Effect: Prediction that non-
nucleonic component in the deuteron wave function may result in angular
dependence of LF-momentum distribution can be verified at CM momenta $k\gtrsim
1$ GeV/c. This seems incredibly large momenta to be measured in experiment.
However the first such measurement at high $Q^{2}$ disintegration of the
deuteron is already performed at Jefferson LabHallC:2020kdm  reaching $k\sim
1$ GeV/c. It is intriguing that the results of this measurement qualitatively
disagree with predictions based on conventional deuteron wave function once
$k\gtrsim 800$ MeV/C. The planned new measurementBoeglin:2014aca  will
significantly improve the quality of the data allowing possible verification
of the effects discussed in this work. It is worth mentioning that the
analysis of the experiment will require careful account for competing nuclear
effects such as final state interaction for which there is a significant
theoretical and experimental progress during the last decadeSargsian:2009hf ;
Boeglin:2015cha . If the experiment will not find the angular dependence in
the momentum distribution this will allow to set a new limit on the dominance
of $pn$ component at instantaneous high nuclear densities that corresponds to
$\sim 1$ GeV/c internal momentum in the deuteron. If, however, the angular
dependence is found, it will motivate theoretical modeling of non-nucleonic
components in the deuteron, such as $\Delta\Delta$, $N^{*}N$ or hidden-color
$N_{c}N_{c}$ that can reproduce the observed result. In both cases results of
such studies will advance the understanding of the dynamics of high density
nuclear matter and the relevance of the quark-hadron transition. Possibility
of studies of tensor asymmetries will significantly complement above studies,
however feasibility of such experiments currently is not clear.

Acknowledgments: We are thankful to Drs. Leonid Frankfurt and Mark Strikman
for useful discussions. This work is supported by the U.S. DOE Office of
Nuclear Physics grant DE-FG02-01ER41172.

## References

  * (1) J. J. Aubert et al. [European Muon], Phys. Lett. B 123, 275-278 (1983).
  * (2) L. L. Frankfurt and M. I. Strikman, Phys. Rept. 160, 235-427 (1988).
  * (3) O. Hen, G. A. Miller, E. Piasetzky and L. B. Weinstein, Rev. Mod. Phys. 89, no.4, 045002 (2017).
  * (4) J. Seely, et al. Phys. Rev. Lett. 103, 202301 (2009).
  * (5) L. B. Weinstein, et al. Phys. Rev. Lett. 106, 052301 (2011).
  * (6) B. Schmookler et al. [CLAS], Nature 566, no.7744, 354-358 (2019).
  * (7) E. Fonseca, et al. Astrophys. J. Lett. 915, no.1, L12 (2021).
  * (8) M. C. Miller, et al. Astrophys. J. Lett. 918, no.2, L28 (2021).
  * (9) W. Melnitchouk, M. Sargsian and M. I. Strikman, Z. Phys. A 359, 99-109 (1997).
  * (10) R. P. Feynman, “Photon-hadron interactions, ”CRC Press (January 1, 1972).
  * (11) L. Frankfurt and M. Strikman, Phys. Rept. 76, 215-347 (1981).
  * (12) G. A. Miller, Prog. Part. Nucl. Phys. 45, 83-155 (2000).
  * (13) S. J. Brodsky, H. C. Pauli and S. S. Pinsky, Phys. Rept. 301, 299-486 (1998).
  * (14) T. Frederico, E. M. Henley and G. A. Miller, Nucl. Phys. A 533, 617-641 (1991).
  * (15) F. Vera, [arXiv:2108.11502 [nucl-th]].
  * (16) M.M. Sargsian and F. Vera, in progress
  * (17) G. P. Lepage and S. J. Brodsky, Phys. Rev. D 22, 2157 (1980).
  * (18) J. Carbonell and V. A. Karmanov, Nucl. Phys. A 581, 625-653 (1995).
  * (19) S. Weinberg, Phys. Rev. 150, 1313-1318 (1966).
  * (20) W. Buck and F. Gross, Phys. Rev. D 20, 2361 (1979).
  * (21) J. Carbonell, B. Desplanques, V. A. Karmanov and J. F. Mathiot, Phys. Rept. 300, 215-347 (1998).
  * (22) L. L. Frankfurt and M. I. Strikman, in Modern topics in electron scattering, edited by B. Frois and I. Sick, 1991.
  * (23) L.L. Frankfurt, M.I. Strikman, L. Mankiewicz, and M. Sawicki, Few-Body Systems 8, 37-43 (1990).
  * (24) P. V. Degtyarenko, Y. V. Efremenko, V. B. Gavrilov and G. A. Leksin, Z. Phys. A 335, 231-238 (1990).
  * (25) C. Yero et al. [Hall C], Phys. Rev. Lett. 125, no.26, 262501 (2020).
  * (26) W. U. Boeglin, et al. [arXiv:1410.6770 [nucl-ex]].
  * (27) M. M. Sargsian, Phys. Rev. C 82, 014612 (2010).
  * (28) W. Boeglin and M. Sargsian, Int. J. Mod. Phys. E 24, no.03, 1530003 (2015).