Artinian bimodule with quasi-Frobenius bimodule of translations

TL;DR

This paper explores the relationship between Artinian bimodules with quasi-Frobenius properties and their translation bimodules, focusing on the structure of associated multiplication and center rings.

Contribution

It establishes new connections between quasi-Frobenius properties of (A,B)-bimodules and their translation (C,Z)-bimodules over Artinian rings.

Findings

Quasi-Frobenius property transfers under certain conditions.

Characterization of translation bimodules in the Artinian setting.

Structural insights into multiplication and center rings.

Abstract

Let M be an (A,B)-bi-module over left and right respectively Artinian rings. Let C be a multiplication ring of bi-module M, i.e. the ring C is generated by images of a and B in End(M). Let Z be a center of C. Bi-module M over rings C and Z we call a bi-module of translation of (A,B)-bi-module M. We investigate a relationship between the fact that (A,B)-bi-module M is quasi-Frobenius and the fact that (C,Z)-bi-module M is quasi-Frobenius.

Equations

$$e=\sum _{l=1}^t\sum _{i=1}^{k_l}{e}_{l,i},$$

$$A=\sum _{l=1}^t\oplus \sum _{i=1}^{k_l}\oplus A{e}_{l,i}$$

$$A=\sum _{l=1}^t\oplus A_l\oplus \mathcal{U},$$

$$J(A)=\sum _{l=1}^{t}J(A_l)\oplus \mathcal{U}.$$

$$\begin{array}[]{cc}I=l_{A}(r_{M}(I)),&W=r_{M}(l_{A}(W)),\\ \end{array}$$

$$\begin{array}[]{cc}J=r_{B}(l_{M}(I)),&V=l_{M}(r_{B}(V)).\end{array}$$

$$N=r_M(I),J=r_B(N);$$

$$N=l_M(J),I=l_A(N).$$

$$\mathfrak{S}(M)=\mathfrak{S}({}_{A}M)=\mathfrak{S}(M_B)$$

$${}_A{M}_B={}_{S_{k,k}}{W_{k,l}}_{T_{l,l}},$$

$$M_Z\cong Q_Z^{(n)},$$

$$C\cong D_{n,n},$$

$$C\cong Z_{n,n}.$$

$${}_C{M}_Z\cong {}_{Z_{n,n}}{Q}_Z^{(n)}.$$

$$(\hat{a}c{)}^{n(\hat{A})}=\sum _{\left(i_1,\dots ,i_{n(\hat{A})}\right)}\hat{a}{\hat{a}}_{i_1}\cdots \hat{a}{\hat{a}}_{i_{n(\hat{A})}}\cdot {\check{b}}_{i_1}\cdots {\check{b}}_{n(\hat{A})},$$

$$\hat{a}{\hat{a}}_{i_1}\cdots \hat{a}{\hat{a}}_{i_{n(\hat{A})}}\in J(\hat{A}{)}^{n(\hat{A})}=\hat{0},$$

$$J(\hat{A})=J(C)\cap J(\hat{A})=J(C)\cap \hat{A}$$

$$J(\check{B})=J(C)\cap J(\check{B})=J(C)\cap \check{B}.$$

$${}_{\bar{C}}\mathfrak{S}(M{)}_{\bar{Z}}\cong {}_{{\bar{Z}}_{n,n}}{\bar{Z}}_{\bar{Z}}^{(n)}.$$

$$\omega (C)=[\omega (\hat{A}),\omega (\check{B})],$$

$$\omega (\hat{A})\cong \hat{A}/(\hat{A}\cap J(C))=\hat{A}/J(\hat{A})=\bar{\hat{A}}\cong \bar{A}$$

$$\omega (\check{B})\cong \check{B}/(\check{B}\cap J(C))=\check{B}/J(\check{B})=\overline{\stackrel{ˇ}{B}}\cong {\bar{B}}^{op}.$$

$$\bar{C}=[\omega (\hat{A}),\omega (\check{B})]=\sum _{\genfrac{}{}{0ex}{}{i=\overline{1,t}}{j=\overline{1,l}}}\oplus [\omega ({\hat{A}}_i),\omega ({\check{B}}_j)].$$

$$\bar{Z}=Z(\bar{C})<Z(\omega (\hat{A})),Z(\omega (\check{B}))<Z(\bar{C})=\bar{Z}.$$

$$\bar{C}\cong (\Delta \underset{\bar{Z}}{\otimes }\nabla {)}_{rs,rs}.$$

$$\Delta \underset{\bar{Z}}{\otimes }\nabla \cong \Delta \underset{\bar{Z}}{\otimes }{\Delta }^{op}\cong {\bar{Z}}_{u,u},$$

$${\dim }_{\bar{Z}}\mathfrak{S}(M)=r\cdot s\cdot {\dim }_{\bar{Z}}\Delta .$$

$$\mathfrak{S}(M)=\sum _{\genfrac{}{}{0ex}{}{i=\overline{1,r}}{u=\overline{1,s}}}\oplus e_{i,i}\mathfrak{S}(M)f_{u,u}.$$

$$e_{i,i}\omega (\hat{A})e_{i,i}\cong \Delta ,\omega (\hat{A})\cong {\Delta }_{r,r},$$

$$f_{u,u}\omega (\check{B}{)}^{op}{f}_{u,u}\cong {\nabla }^{op}\cong \Delta ,\omega (\check{B}{)}^{op}\cong {\Delta }_{s,s},$$

Full text

We investigate mutual relationship of two cases :

(а) bimodule ${}_{A}M_{B}$ is a Morita’s Artinian duality context and

(b) bimodule ${}_{C}M_{Z}$ is a Morita’s Artinian duality context.

Let’s note that sometimes bimodule of translations is called * canonical bimodule * and $C$ is called *multiplication * ring.

1 Introduction

Main results of this article previously were presented at Workshop [24]. Area of this investigation had arose in 1990-s when A.A.Nechaev рas had attempts to generalize the notion of linear recurrent sequence over commutative ring to the cases of linear recurrences over non-commutative ring, module, bimodule [19, 20, 21, 14, 22].

Let’s note immediately that the necessity for the ring, module, bimodule to be quasi-Frobenius [2] was established promptly. Further one of the possible ways to determine linear recurrences over non-commutative ring, module or bimodule was this one [23].

Let ${}_{A}M_{B}$ be an arbitrary bimodule.

• By the left translation * [23] generated by the element $a\in A$ is called a natural map $\hat{a}:M\to M$ defined by equality : $\forall m\in M$ $\hat{a}(m)=am$.
• Analogously the right translation * [23] generated by the element $b\in B$ is a natural map $\check{b}$ defined by the equality : $\forall m\in M$ $\check{b}(m)=mb$. It is evidently that $\hat{a}\in\mathop{\rm End}(M)=\mathop{\rm End}(M_{{\mathbb{Z}}})$ and $\check{b}\in\mathop{\rm End}(M)$. Subring $\hat{A}=\{\hat{a}\;|\;a\in A\}$ of the ring $\mathop{\rm End}(M)$ is called * ring of left translations.* Respectively ring $\check{B}=\{\check{b}\;|\;b\in B\}$ is called
• ring of right translations.*

Let’s remember that rings $(K,+,*)$ and $(R,\oplus,\otimes)$ are called * inversely isomorphic or anti-isomorphic * [7], if ring’s isomorphism takes place: $K^{op}\cong R$ where $K^{op}=(K,+,\tilde{*})$ and operation $\tilde{*}:K\times K\to K$ is defined according to the rule: $k_{1}\tilde{*}k_{2}=k_{2}*k_{1},\ k_{1},k_{2}\in K$.

It is easy to see that if ${}_{A}M$ is a faithful module then rings $A$ and $\hat{A}$ are isomorphic and if $M_{B}$ is a faithful module then rings $B$ and $\check{B}$ are inversely isomorphic .

Lemma 1**.**

For arbitrary bimodule ${}_{A}M_{B}$ this equality takes place: $\hat{A}\cap\check{B}=\mathop{\rm\bf Z}(\hat{A})\cap\mathop{\rm\bf Z}(\check{B})$. Here $\mathop{\rm\bf Z}(\hat{A})$ and $\mathop{\rm\bf Z}(\check{B})$ denote centers of corresponding rings.

Proof.

Because of the associativity $(am)b=a(mb)$ elements of the rings $\hat{A}$ and $\check{B}$ are pair-wise commutative. Hence arbitrary element $z\in\hat{A}\cap\check{B}$ lies in $\mathop{\rm\bf Z}(\hat{A})$ and in $\mathop{\rm\bf Z}(\check{B})$ concurrently, i.e. $\hat{A}\cap\check{B}\subset\mathop{\rm\bf Z}(\hat{A})\cap\mathop{\rm\bf Z}(\check{B})$. Inverse inclusion is evident. ∎

Commutative ring $Z=\hat{A}\cap\check{B}$ ( Lemma 1) is called [23]

• common center of rings $A$ and $B$ in relation to bimodule $M$. * Let’s denote by $\phi:\hat{A}\to A$ isomorphism constructed according to the rule: $\hat{a}\stackrel{{\scriptstyle\phi}}{{\mapsto}}a$ and by $\psi:\check{B}\to B$ inverse isomorphism of rings $\check{B}$ and $B$ constructed by the rule: $\check{b}\stackrel{{\scriptstyle\psi}}{{\mapsto}}b$. Then $\phi(Z)<\mathop{\rm\bf Z}(A)$, $\psi(Z)<\mathop{\rm\bf Z}(B)$.

Besides that to the set $M$ may be correctly assigned structure of $(Z,Z)$-bimodule in such way that these identities hold: $z\cdot m=\phi(z)m$, $m\cdot z=m\psi(z)$, $zm=mz$.

$(Z,Z)$-bimodule $\hat{A}\mathop{\otimes}\limits_{Z}\check{B}$ has an associative multiplication operator [25, Proposition 9.2(a)] defined by the rule: $(\hat{a}_{1}\otimes\check{b}_{1})(\hat{a}_{2}\otimes\check{b}_{2})=\hat{a}_{1}\hat{a}_{2}\otimes\check{b}_{1}\check{b}_{2}$. Herewith if $A$ and $B$ are rings with identity then $1_{\hat{A}\otimes\check{B}}=1_{\hat{A}}\otimes 1_{\check{B}}$. Now $\hat{A}\mathop{\otimes}\limits_{Z}\check{B}$ is a ring and $M$ is a [25, Proposition 10.1(a)] left $\hat{A}\mathop{\otimes}\limits_{Z}\check{B}$-module with multiplication according to the rule: $(\hat{a}\otimes\check{b})m=\phi(\hat{a})m\psi(\check{b})$. If modules ${}_{A}M$ и $M_{B}$ are unitary then $1_{\hat{A}\otimes\check{B}}=\epsilon$ is an identity mapping of $M$.

Further we will need this generalization of [25, $\S 10.1$, Proposition ]:

Proposition 2**.**

Let $A,B$ are $Z$-algebras. Any $(A,B)$-bimodule $M$ is a left $A\mathop{\otimes}\limits_{Z}B^{op}$-module with scalar multiplication defined according to the rule: $(x\otimes y)u=(xu)y=x(uy)$ for $x\in A,y\in B$, $u\in M$.

Conversely, any left $A\otimes B^{op}$-module is an $(A,B)$-bimodule with operations: $xu=(x\otimes 1_{B})u,\ \ \ uy=(1_{A}\otimes y)u.$ If $M$ and $N$ are $(A,B)$-bimodules then $\mathop{\rm Hom}_{A-B}(M,N)=\mathop{\rm Hom}_{A\otimes B^{op}}({}_{A\otimes B^{op}}M,{}_{A\otimes B^{op}}N).$

Ring $C$ is called * ring of translations of bimodule ${}_{A}M_{B}$* [23] if it is generated in the ring $\mathop{\rm End}(M)$ by the union of rings $\hat{A}$ and $\check{B}$.

It takes place

Statement 3**.**

Mapping $\lambda:\hat{A}\mathop{\otimes}\limits_{Z}\check{B}\to C$ satisfying to the rule: $\lambda\left(\sum\limits_{(i)}\hat{a}_{i}\otimes\check{b}_{i}\right)=\sum\limits_{(i)}\hat{a}_{i}\check{b}_{i}$ is an epimorphism of rings and $\ker\lambda=\{\delta\in\hat{A}\mathop{\otimes}\limits_{Z}\check{B}\ |\ \delta M=\theta\}$ where $\theta$ is an identity element of $M$.

Proof.

The first statement follows from element-wise commutativity of rings $\hat{A}$ and $\check{B}$. The second statement follows from the fact that ${}_{C}M$ is a faithful module. ∎

Previously [23] bimodule ${}_{C}M_{Z}$ was called

• canonical bimodule of bimodule ${}_{A}M_{B}$. *

Further we instead or denomination * canonical bimodule * will use denomination

• bimodule of translations *.

Now *linear recurrent sequence * over bimodule ${}_{A}M_{B}$ may be defined in such way [23]: it is a mapping $\mu:{\mathbb{N}}_{0}\to M$ for which there exists a unitary polynomial $c(x)=x^{m}-\sum_{j=0}^{m-1}c_{j}x^{j}\in C[x]$ with property: $\forall i\in{\mathbb{N}}_{0}\ \mu(i+m)=\sum_{j=0}^{m-1}c_{j}\left(\mu(i+j)\right)$.

Herewith to generalize classical properties of linear recurrences onto newly arisen case it is necessary for bimodule ${}_{C}M_{Z}$ to be quasi-Frobenius.

Let’s remember that according to definition [2] left-faithful and right-faithful bimodule $M$ over rings $A$ and $B$ with identities is called quasi-Frobenius if for every maximal left ideal $I$ of ring $A$ *right annihilator * $r_{M}(I)=\{m\in M\;|\;Im=\theta\}$ is either irreducible $B$-module either equal to neutral element of $M$ and for every maximal right ideal $J$ of ring $B$ *left annihilator * $l_{M}(J)=\{m\in M\;|\;mJ=\theta\}$ is either irreducible $A$-module either equal to $\theta$. Here and after $\theta$ is a neutral element of the group $(M,+)$.

However it is a problem that there were not known any facts about relations between quasifrobeniusness of bimodule ${}_{C}M_{Z}$ and qusifrobeniusness of bimodule ${}_{A}M_{B}$. Initially was formulated

Hypothesis 4** ([23]).**

If bimodule ${}_{A}M_{B}$ is a Morita’s Artinian duality context then bimodule of translations ${}_{C}M_{Z}$ is also a Morita’s Artinian duality context.

However during multiyear process of attempts to prove this Hypothesis it become clear that it’s in general case wrong (Theorem 6) and moreover under additional conditions inverse implication holds (Theorem 5).

Resulting in these facts approach to definition of linear recurrences over bimodules suggested by V.L.Kurakin was adopted [9]. Theoretically this approach was based on results of [11].

Despite this results of this article are not negligible and were continued: notion of matrix linear recurrences is based on the Theorem 5 [10, 27] and the notion of skew linear recurrences is based on Theorem 6 [5].

Let’s now get to outline and prove main results of this article. In order of independence and completeness we will remember necessary definitions [7]. Module ${}_{A}M$ is called * injective * if for every monomorphism $\alpha:{}_{A}K\to{}_{A}L$ and every homomorphism $\phi:{}_{A}K\to{}_{A}M$ there exists homomorphism $\psi:{}_{A}L\to{}_{A}M$ with property $\phi=\psi\alpha$. It is equivalent [7, Theorem 5.3.1(a)] to the fact that for every monomorphism $\xi:{}_{A}M\to{}_{A}N$ module $\xi(M)$ allocated in $N$ as a direct summand. According to [7, Theorem 5.6.4] for every $A$-module $L$ there exists unique up to isomorphism * injective hull * $\mathop{\rm\bf I}(L)$ i.e. minimal according to order of inclusion injective module containing $L$. Module ${}_{A}U>{}_{A}L$ is called * essential extension * of module ${}_{A}L$ if for every submodule $V$ of module ${}_{A}U$ from condition $L\cap V=\theta$ follows that $V=\theta$. It is known [7, Теорема 5.6.6] that injective hull $\mathop{\rm\bf I}(L)$ of module $L$ is a maximal essential extension of $L$.

Also we have to remember the notion of * Brauer group of the field $F$* [25, $\S 12.5$].

Let $F$ be a field. Let’s denote by ${\mathfrak{B}}(F)$ class of all finite-dimensional simple $F$-algebras $A$ with property $\mathop{\rm\bf Z}(A)=F$ i.e. class of all finite-dimensional central simple $F$-algebras.

If algebras $A$ and $B$ contains in ${\mathfrak{B}}(F)$ then [25, $\S 12.4$, Proposition b(i)] $F$-algebra $A\mathop{\otimes}\limits_{F}B$ contains in ${\mathfrak{B}}(F)$.

*Besides that [25, $\S 12.5$, Lemma ] following conditions are equivalent: *

(i) basic algebras of $A$ и $B$ are isomorphic;

(ii) there exists an algebra with division (body) $D\in{\mathfrak{B}}(F)$ and naturals $m$ and $n$ such that $A\cong D_{n,n}$ and $B\cong D_{m,m}$;

*(iii) there exists naturals $r$ and $s$ such that $A\otimes F_{r,r}\cong B\otimes F_{s,s}$. *

Algebras $A,B\in{\mathfrak{B}}(F)$, underlying this conditions are called

• equivalent. * Equivalence class of algebra $A$ is denoted as $[A]$.

The set $\mathop{\rm\bf B}(F)=\{[A]\ |\ A\in{\mathfrak{B}}(F)\}$ is [25, $\S 12.5$, Proposition a] an Abel group with operator $[A][B]=[A\otimes B]$, neutral element $[F]$ and the operation of taking inverse element $[A]^{-1}=[A^{op}]$. The group $\mathop{\rm\bf B}(F)$ is called Brauer group * of the field $F$. * Every class in group $\mathop{\rm\bf B}(F)$ is represented by [25, $\S 12.5$, Proposition b(ii)] algebra with division which is unique up to isomorphism . If Brauer group of field $F$ is trivial then every central simple $F$-algebra is a full matrix algebra over $F$.

• If the field $F$ is algebraically closed then its Brauer group is trivial * [25, $\S 12.5$, Corollary]. Besides that Brauer group of the finite field is trivial too.

By * socle * ${\mathfrak{S}}({}_{A}M)$ of the left $A$-module $M$ is called [6, Chapter IV] the sum of all irreducible submodules of the $A$-module $M$.

The main result of this work consists in following:

Theorem 5**.**

Let bimodule ${}_{A}M_{B}$ be a faithful as a left $A$-module and as a right $B$-module simultaneously, bimodule of translations ${}_{C}M_{Z}$ of bimodule ${}_{A}M_{B}$ is quasi-Frobenius, $Z$ is a local Artinian ring, $Q=\mathop{\rm\bf I}(\bar{Z}_{Z})$ is an injective hull of the unique irreducible $Z$-module $\bar{Z}$.

Then:

I. (a) Ring’s isomorphism takes place: $C\cong Z_{n,n}$.

(b) Bimodule’s isomorphism takes place: ${}_{C}M_{Z}\cong{}_{Z_{n,n}}Q^{(n)}_{Z}$.

(c) Rings $A$, $B$ are primary Artinian left-side and right-side simultaneously rings with identity and factor-rings $\bar{A}=A/J(A)$, $\bar{B}=B/J(B)$ of rings $A$ and $B$ respectively when activated modulo Jacobson’s radical are equivalent elements of the set ${\mathfrak{B}}(\bar{Z})$.

(d) Bimodule ${}_{\bar{A}}{\mathfrak{S}}({}_{C}M)_{\bar{B}}$ is quasi-Frobenius.

II. Bimodule ${}_{A}M_{B}$ is quasi-Frobenius if and only if the left socle ${\mathfrak{S}}({}_{A}M)$ and the right socle ${\mathfrak{S}}(M_{B})$ of bimodule ${}_{A}M_{B}$ are equal to ${\mathfrak{S}}({}_{C}M)$.

III. Under additional condition of the form: Brauer group $\mathop{\rm\bf B}(\bar{Z})$ of the field $\bar{Z}$ is trivial bimodule ${}_{A}M_{B}$ is quasi-Frobenius and moreover for some naturals $r,s\in{\mathbb{N}}$ such that $n=r\cdot s$ isomorphisms of rings and bimodules respectively takes place:

(a) $A\cong Z_{r,r}$, $B\cong Z_{s,s}$,

(b) ${}_{A}M_{B}\cong{}_{Z_{r,r}}{Q_{r,s}}_{Z_{s,s}}$.

Here we denote by ${}_{Z_{r,r}}{Q_{r,s}}_{Z_{s,s}}$ the set of orthogonal matrices of dimensions $r\times s$ filled by elements of the set $Q$ and usual matrix multiplication upon elements of $Z_{r,r}$ and $Z_{s,s}$ respectively. It is evident that ${}_{Z_{r,r}}{Q_{r,s}}_{Z_{s,s}}$ is a bimodule.

Conversion of the point III of Theorem 5 is wrong even in the finite case. Appropriate example may be constructed in class of so called $GEO$-rings [18]. By definition the ring $S$ is called * Galois-Eisenstein-Ore ring * (or * $GEO$-ring *) if it is finite completely primary ring of principal ideals i.e. the ring $S$ contains unique (one-side) maximal ideal ${\mathfrak{p}}(S)$ and every one-side ideal of ring $S$ is principal. Commutative $GEO$-ring is called Galois-Eisenstein ring or $GE$-ring [18].

In arbitrary $GEO$-ring $S$ of characteristic $p^{d}$ contains

• Galois subring * $R=GR(p^{d},r)$ [13, 26], $q=p^{r}$. All such rings are conjugated in $S$. Ring $R$ is called *coefficients ring * of $S$.

Theorem 6**.**

(a) For arbitrary $GEO$-ring $S$ bimodule ${}_{S}S_{S}$ is a quasi-Frobenius.

(b) Let $S$ be a $GEO$-ring with a coefficients ring $R=GR(p^{d},r)$ and in addition $d>1$ and $(r,p)=1$. Then bimodule of translations ${}_{C}S_{Z}$ of quasi-Frobenius $S$-bimodule $S$ is a quasi-Frobenius bimodule if and only if $S$ is a $GE$-ring.

2 Preliminaries

To prove Theorems above it is necessary to remember a lot of definitions and facts connected firstly with different characterizations of quasi-Frobenius bimodules. Most deeply and fully quasi-Frobenius bimodules are investigated in the case when rings $A$ and $B$ are *Artinian * left-side and right-side respectively, i.e. it satisfy descending chain conditions of left (respectively, right) ideals. Such quasi-Frobenius bimodules in the monography [4] are called

• (Morita’s) Artinian duality context *. In this case [2, Theorem 6(iv),(v)] modules ${}_{A}M$ and $M_{B}$ are finitely generated.

Let’s remember other well-known results about quasi-Frobenius bimodules.

Let $A$ be a left Artinian ring with unit $e$, $J(A)$ be a Jacobson radical of ring $A$. Then $J(A)$ ([6, Corollary III.1.1]) is a maximal nil-potent two-sided ideal of ring $A$ containing every one-side nil-ideal of ring $A$. It is known [6, Chapter III] that $\bar{A}=A/J(A)$ is a direct sum of pair-wise orthogonal simple rings $\bar{A}_{1},\ldots,\bar{A}_{t}$ with units $\bar{e}^{(1)},\ldots,\bar{e}^{(t)}$ respectively. Idempotent $\bar{e}_{l}$, $l\in\overline{1,t}$ of ring $\bar{A}_{l}$ is called [6] * primitive * if left ideal $\bar{A}\bar{e}_{l}$ of ring $\bar{A}$ is irreducible as a left $\bar{A}$-module. Then $\Delta_{l}=\mathop{\rm End}\left({}_{\bar{A}}\bar{A}\bar{e}_{l}\right)$ is a body isomorphic to the ring $\bar{e}_{l}\bar{A}\bar{e}_{l}=\bar{e}_{l}A_{l}\bar{e}_{l}$ and ring $\bar{A}_{l}$ is a full matrix algebra of finite degree $k_{l}$ over body $\Delta_{l}$. Besides that modules $\bar{A}_{l}\bar{e}_{l}$,$l=\overline{1,t}$ are all, up to isomorphism, irreducible $A$-modules.

Analogously idempotent $e_{l}$, $l\in\overline{1,t}$, of ring $A$ is called [6] *primitive * if left ideal $\bar{A}\bar{e}_{l}$ of ring $\bar{A}$ is irreducible left $\bar{A}$-module.

Idempotents $\bar{e}_{j},\bar{e}_{l}\in\bar{A}$ are called isomorphic if isomorphic left $\bar{A}$-modules $\bar{A}\bar{e}_{j}$ and $\bar{A}\bar{e}_{l}$. Analogously idempotents $e_{j},e_{l}\in A$ are called isomorphic if isomorphic left $A$-modules $Ae_{j}$ and $Ae_{l}$.

Let $\bar{e}_{l}$, $l=\overline{1,t}$, are all non-isomorphic idempotents of ring $\bar{A}$.

For every $l=\overline{1,t}$ there exists idempotent $e_{l}\in\bar{e}_{l}$, such that elements $e_{l}$, $l=\overline{1,t}$, are all non-isomorphic primitive idempotents of ring $A$, and there takes place [6, Proposition III.8.5] the expansion unit $e$ of ring $A$ into the sum of primitive orthogonal idempotents

[TABLE]

where every idempotent $e_{l,i}$ is isomorphic to idempotent $e_{l}$ (i.e. by definition ${}_{A}Ae_{l,i}\cong{}_{A}Ae_{l}$).

Decomposition (2.1) generates decomposition of the ring $A$ into the direct sum of undecomposable left ideals:

[TABLE]

Let’s denote $e^{(l)}=\sum\limits_{i=1}^{k_{l}}e_{l,i}$, $l=\overline{1,t}$. It is known [6, Theorem III.9.2] that additive group of the ring $A$ is represented as a direct sum of Abel groups

[TABLE]

where $A_{l}=e^{(l)}Ae^{(l)}$, $l=\overline{1,t}$, ${\cal U}=\sum\limits_{l\neq\nu}e^{(l)}Ae^{(\nu)}$. Herewith $A_{l}=S^{(l)}_{k_{l},k_{l}}$, $l=\overline{1,t}$ are primarily rings with identities $e^{(l)}$ i.e. every of it is a full matrix ring over local left Artinian ring $S^{(l)}$ wherein it’s a preimage of ring $\bar{A}_{l}$ when activating modulo $J(A)$, ${\cal U}$ is a subgroup of group $(J(A),+)$. Moreover herewith

[TABLE]

Further rings $A_{l}$, $l=\overline{1,t}$, we will call *primarily components * of ring $A$. If $t=1$ then [6, Theorem III.8.1] $A=S_{k,k}$ is a primary ring.

Let’s remember that socle ${\mathfrak{S}}({}_{A}M)$ of the left $A$-module $M$ [6, Chapter IV] is a completely reducible left $A$-module satisfying condition: ${\mathfrak{S}}({}_{A}M)=r_{M}(J(A))$. Sum ${\mathfrak{S}}^{(V)}({}_{A}M)=\sum_{i\in I}\tilde{V}_{i}$ of all irreducible submodules $\tilde{V}_{i},i\in I$, of module ${}_{A}M$ isomorphic to module ${}_{A}V$ is called

• homogeneous component of socle ${\mathfrak{S}}({}_{A}M)$ belonging to irreducible left $A$-module $V$.

If ${}_{A}M_{B}$ is some bimodule and $V$ is a some left $A$-module then $B$-module $V^{*}_{B}=\mathop{\rm Hom}\nolimits_{A}(V,M)$ is called * a right dual to $V$ with respect to bimodule $M$* [6, Chapter IV]. Relatively by left dual to the right $B$-module $W$ with respect to ${}_{A}M_{B}$ is called $A$-module ${}_{A}W^{*}=\mathop{\rm Hom}\nolimits_{B}(W,M)$.

If ${}_{A}M_{B}$ is a quasi-Frobenius bimodule, ${}_{A}V$ and $W_{B}$ are irreducible modules then $V_{B}^{*}$ и ${}_{A}W^{*}$ are also irreducible (either are non-zero). Moreover [2, Theorem 1] these equalities take place: ${\mathfrak{S}}^{(V)}({}_{A}M)={\mathfrak{S}}^{(V^{*})}(M_{B})$, ${\mathfrak{S}}^{(W)}(M_{B})={\mathfrak{S}}^{(W^{*})}({}_{A}M)$, where $V^{*}_{B}=\mathop{\rm Hom}\nolimits_{A}(V,M)$, ${}_{A}W^{*}=\mathop{\rm Hom}\nolimits_{B}(W,M)$. Hence ${\mathfrak{S}}({}_{A}M)={\mathfrak{S}}(M_{B})$.

Left $A$-module $M$ is called * distinguished [2] * if it contains an isomorphic image of every irreducible $A$-module or, what’s equivalent, for every irreducible left $A$-module $V$ ${\mathfrak{S}}^{(V)}({}_{A}M)\neq\theta$. Left $A$-module $U$ is called

• minimal (in $M$) [2] * if every submodule of ${}_{A}M$ contains $U$. If minimal submodule exists it is a unique irreducible submodule of ${}_{A}M$.

Now we can provide necessary characterizations of quasi-Frobenius bimodules:

Theorem 7**.**

Following conditions are equivalent:

(1) [2, Theorem 6(iii)] Bimodule ${}_{A}M_{B}$ is a quasi-Frobenius, rings $A$ and $B$ are left Artinian and right Artinian respectively.

(2) [2, Theorem 6(i)] Ring $A$ is a left Artinian, left $A$-module $M$ is an injective, finitely generated and distinguished, ring $\check{B}$ is an endomorphism ring of module ${}_{A}M$.

*(3)[4, Theorem 23.25(6)] Every left ideal $I$ of ring $A$, every right $B$-submodule $W$ of module $M$ are satisfies to

• **annihilator correspondences: ****

[TABLE]

*every right ideal $J$ of ring $B$, every left $A$-submodule of $M$ satisfy to * **annihilator correspondences: ****

[TABLE]

(4) [2, Theorem 12] Bimodule ${}_{A}M_{B}$ is a faithful, $A$ is a left Artinian ring, $B$ is a right Artinian ring, $A$ and $B$ have the same number $t$ of non-isomorphic primitive idempotents, namely: $e_{1},\ldots,e_{t}$, $f_{1},\ldots,f_{t}$ respectively, which under necessary re-enumeration satisfy conditions:

I. For every $l\in\overline{1,t}$ left $A$-module $Mf_{l}$ contains minimal $A$-submodule isomorphic to $\bar{A}\bar{e}_{l}$, an image of $Ae_{l}$ activating modulo $J(A)e_{l}$;

II. For every $l\in\overline{1,t}$ right $B$-module $e_{l}M$ contains minimal $B$-submodule isomorphic to $\bar{f}_{l}\bar{B}$, an image of $f_{l}B$ activating modulo $f_{l}J(B)$;

(5) [2, Theorem 6(ii)] Ring $B$ is right Artinian, right $B$-module $M$ is injective, finitely generated and distinguished, ring $\hat{A}$ is an endomorphism ring of module $M_{B}$.

If ${}_{A}M_{B}$ is a quasi-Frobenius bimodule satisfying to conditions of Theorem 7 then [2, Theorem 7] there exist one-to-one Galois correspondences between the set of two-sided ideals $I$ of ring $A$, the set of $(A,B)$-subbimodules $N$ of bimodule $M$, and the set of two-sided ideals $J$ of ring $B$, which are established by equalities:

[TABLE]

[TABLE]

Herewith $N$ is a quasi-Frobenius $(A/I,B/J)$-bimodule. In particular, ${\mathfrak{S}}(M)$ is a quasi-Frobenius $(A/J(A),B/J(B))$-bimodule.

Further we will use this

Statement 8** (A.A.Nechaev, 1996г.).**

Left-faithful and right-faithful bimodule ${}_{A}M_{B}$ over left Artinian and right Artinian correspondingly rings with units is quasi-Frobenius if and only if left socle ${\mathfrak{S}}({}_{A}M)$ and right socle ${\mathfrak{S}}(M_{B})$ of $M$ coincides and $(\bar{A},\bar{B})$-bimodule

[TABLE]

is quasi-Frobenius.

Proof.

For quasi-Frobenius bimodule ${}_{A}M_{B}$ we have equality: ${\mathfrak{S}}({}_{A}M)={\mathfrak{S}}(M_{B})$ [2, Theorem 1]. Because ${}_{A}M_{B}$ is satisfying to conditions of Theorem 7 we have that [2, Theorem 7] ${\mathfrak{S}}(M)$ is a quasi-Frobenius $(\bar{A},\bar{B})$-bimodule.

Conversely let equalities (2.7) take place and bimodule ${}_{\bar{A}}{\mathfrak{S}}(M)_{\bar{B}}$ is a quasi-Frobenius. Then bimodule ${}_{\bar{A}}{\mathfrak{S}}(M)_{\bar{B}}$ satisfies conditions of point 4 of Theorem 7 and according to it notations these equalities take place: ${\mathfrak{S}}(e_{l}M_{B})={\mathfrak{S}}(\bar{e}_{l}{\mathfrak{S}}(M)_{\bar{B}})$ and ${\mathfrak{S}}({}_{A}Mf_{l})={\mathfrak{S}}({}_{\bar{A}}{\mathfrak{S}}(M)\bar{f}_{l})$. Hence if conditions of point 4 of Theorem 7 take place relatively to bimodule ${}_{\bar{A}}{\mathfrak{S}}(M)_{\bar{B}}$ then it implies satisfying of the same conditions in relation to bimodule ${}_{A}M_{B}$. ∎

Lemma 9**.**

For Artinian duality context ${}_{A}M_{B}$ this equality takes place: $\mathop{\rm\bf Z}(\hat{A})=\mathop{\rm\bf Z}(\check{B})$.

Proof.

According to Theorem 7(2), $\hat{A}=\mathop{\rm End}(M_{B})$. Because an arbitrary element $z\in\mathop{\rm\bf Z}(\check{B})$ contains in $\mathop{\rm End}(M_{B})$ then $\mathop{\rm\bf Z}(\check{B})\subset\hat{A}$ what follows $\mathop{\rm\bf Z}(\check{B})\subset\mathop{\rm\bf Z}(\hat{A})$. Converse inclusion may be stated by the same method. ∎

It is also true

Statement 10**.**

If $M$ is a bimodule over primary left-Artinian and right-Artinian rings with identities respectively $A=S_{k,k}$ and $B=T_{l,l}$ of the form:

[TABLE]

then bimodule ${}_{A}M_{B}$ is a quasi-Frobenius if and only if bimodule ${}_{S}W_{T}$ is a quasi-Frobenius.

Proof.

Let suppose that bimodule ${}_{A}M_{B}$ is a quasi-Frobenius. Let $e$ be a primitive idempotent of ring $A$. Then according to point (4) of Theorem 7 right $B=T_{l,l}$-module $eM=W^{(l)}$ contains a minimal submodule. Let suppose now that right $T$-module $W$ does not contain minimal submodule. It means that module $W_{T}$ contains at least two diffrerent irreducible submodules $U$ and $V$. Then module $W^{(l)}_{T_{l,l}}$ contains at least two different irreducible submodules $U^{(l)}_{T_{l,l}}$ and $V^{(l)}_{T_{l,l}}$. It is a contradiction to existence in $W^{(l)}{}_{T_{l,l}}$ of minimal submodule. Hence right module $W{}_{T}$ also contains minimal submodule.

Analogously may be shown that existence of minimal submodule in ${}_{S_{k,k}}W^{(k)}$ implies existence of minimal submodule in ${}_{S}W$.

Hence according to point (4) of Theorem 7 bimodule ${}_{S}W{}_{T}$ is a quasi-Frobenius.

The converse implication may be proven by the same arguments in reverse order. ∎

3 The Proof of Theorem 5

For convenience of reading let us duplicate the text of Theorem 5.

Theorem 11**.**

Let bimodule ${}_{A}M_{B}$ be faithful as a left $A$-module and as a right $B$-module together, bimodule of translations ${}_{C}M_{Z}$ of ${}_{A}M_{B}$ is a quasi-Frobenius, $Z$ is local Artinian ring, $Q=\mathop{\rm\bf I}(\bar{Z}_{Z})$ is an injective hull of unique irreducible $Z$-module ${}_{Z}\bar{Z}$.

Then:

I. (a) Isomorphism of rings takes place: $C\cong Z_{n,n}$.

(b) Isomorphism of bimodules takes place: ${}_{C}M_{Z}\cong{}_{Z_{n,n}}Q^{(n)}_{Z}$.

(c) Rings $A$, $B$ are primary left-Artinian and right-Artinian rings with identities, herewith factor-rings $\bar{A}=A/J(A)$, $\bar{B}=B/J(B)$ of $A$ and $B$ respectively activated modulo Jacobson radicals are equivalent elements of the set ${\mathfrak{B}}(\bar{Z})$.

(d) Bimodule ${}_{\bar{A}}{\mathfrak{S}}({}_{C}M)_{\bar{B}}$ is a quasi-Frobenius.

II.Bimodule ${}_{A}M_{B}$ is a quasi-Frobenius if and only if the left socle ${\mathfrak{S}}({}_{A}M)$ and the right socle ${\mathfrak{S}}(M_{B})$ of ${}_{A}M_{B}$ coincide with ${\mathfrak{S}}({}_{C}M)$.

III.If additional condition takes place: Brauer group $\mathop{\rm\bf B}(\bar{Z})$ of the field $\bar{Z}$ is trivial, then bimodule ${}_{A}M_{B}$ is a quasi-Frobenius and moreover for some $r,s\in{\mathbb{N}}$ such that $n=r\cdot s$ isomorphisms of rings and bimodules respectively take place:

(a) $A\cong Z_{r,r}$, $B\cong Z_{s,s}$,

(b) ${}_{A}M_{B}\cong{}_{Z_{r,r}}{Q_{r,s}}_{Z_{s,s}}$.

Proof.

I(a). Let ${}_{C}M_{Z}$ be an Artinian duality context. Then according to Theorem 7(5) the right module $M_{Z}$ is an injective and hence it is a direct sum of some $n\in{\mathbb{N}}$ right modules $Q_{Z}$:

[TABLE]

where $Q=\mathop{\rm\bf I}(\bar{Z}_{Z})$ is an injective hull of unique irreducible $Z$-module $\bar{Z}=Z/J(Z)$.

By the same Theorem the equality take place: $\hat{C}=C=\mathop{\rm End}(M_{Z})$. The existence of isomorphism (3.1) implies isomorphism:

[TABLE]

where $D=\mathop{\rm End}(Q_{Z})$. Because of [4, Proposition 23.33], [15], equality takes place: $D=Z$ and isomorphism (3.2) takes a form:

[TABLE]

I(b). From (3.1) and (3.3) follows that

[TABLE]

I(c). From (3.3) follows that $C$ is finitely generated left $Z$-module and finitely generated right $Z$-module simultaneously. Because $Z$ is a commutative Artinian ring then $C$ is also Artinian ring ( and not only left-Artinian how the Theorem 7(1) states ).

Under conditions of this Theorem left module ${}_{A}M$ is a faithful. Hence rings $A$ and $\hat{A}$ are isomorphic. Because ring $\hat{A}$ is a subring of $C$ and hence contains ring $Z$, so module $\hat{A}_{Z}$ is a submodule of the finitely generated module $C_{Z}$ over commutative Artinian ring $Z$. Hence $\hat{A}$ is a two-sided Artinian ring and finitely generated $Z$-module. Because of the isomorphism, ring $A$ is also two-sided Artinian.

Analogously may be established that $\check{B}$ is a two-sided Artinian ring and finitely generated $Z$-module and that ring $B$ is also two-sided Artinian.

Let’s establish now the relation between Jacobson radical $J(C)$ of ring $C$ and Jacobson radicals $J(\hat{A}),J(\check{B})$ of rings $\hat{A}$, $\check{B}$ respectively.

Let $\hat{a}\in J(\hat{A})$. Then right ideal $\hat{a}C$ generated by element $\hat{a}$ in ring $C$ is a nil-ideal. Indeed for arbitrary element $c=\sum\limits_{(i)}\hat{a}_{i}\check{b}_{i}\in C$ because of the element-wise commutativity of rings $\hat{A}$ and $\check{B}$ the equality takes place:

[TABLE]

where $n(\hat{A})$ is a nil-potent index of ideal $J(\hat{A})$. Because for arbitrary set $\left(i_{1},\ldots,i_{n(\hat{A})}\right)$ the inclusion takes place:

[TABLE]

then $(\hat{a}c)^{n(\hat{A})}=\hat{0}$.

Because every one-side nil-ideal of Artinian ring contains in Jacobson radical of it ring we have: $\hat{a}C<J(C)$. Besides that, $\hat{a}\in\hat{a}C$. So we have proved inclusion: $J(\hat{A})\subset J(C)$.

From other side the set $J(C)\cap\hat{A}$ is a two-sided nil-potent ideal of ring $\hat{A}$. Hence, $J(C)\cap\hat{A}\lhd J(\hat{A})$.

So we have a sequence of inclusions: $J(\hat{A})\subset J(C)\cap J(\hat{A})\subset J(C)\cap\hat{A}\lhd J(\hat{A})$ which means that all inclusions are equalities:

[TABLE]

Analogously may be established that

[TABLE]

Let’s denote by ${\mathfrak{S}}(M)$ the socle of the left module ${}_{C}M$ which is equal to [2, Theorem 1] the socle of the right module $M_{Z}$. Because of the [2, Theorem 7] bimodule ${}_{\bar{C}}{\mathfrak{S}}(M)_{\bar{Z}}$ is a quasi-Frobenius. An isomorphism

[TABLE]

may be established by the same arguments as an isomorphism (3.4). From (3.8) follows that $\mathop{\rm\bf Z}(\bar{C})=\bar{Z}$.

Let’s show that rings $\hat{A}$ and $\check{B}$ are primary. Let $\hat{A}_{1},\ldots,\hat{A}_{t}$ and $\check{B}_{1},\ldots,\check{B}_{l}$ are all primary components of rings $\hat{A}$ and $\check{B}$ respectively.

Let $\omega:C\to C/J(C)$ be a canonical epimorphism. Then

[TABLE]

whereas

[TABLE]

and

[TABLE]

Hence because of the [6, Theorem III.9.2]

[TABLE]

Since $\bar{C}\cong{\bar{Z}}_{n,n}$ is a simple ring then between pair-wise orthogonal rings $[\omega(\hat{A}_{i}),\omega(\check{B}_{j})]$, $i=\overline{1,t},j=\overline{1,l}$, there is only one non-zero. Let’s denote it $[\omega(\hat{A}_{1}),\omega(\check{B}_{1})]$, and for all $i,j\geq 2$ inclusions take place: $\hat{A}_{i},\check{B}_{j}<J(C)$. Hence every element of every of rings $\hat{A}_{i},\check{B}_{j}$, $i,j\geq 2$, is a nil-potent what is impossible becaus every of those rings contains an idempotent. It is a contradiction which shows that $t=l=1$ and rings $\hat{A},\check{B}$ are primary.

Let’s note now that because of the pare-wise commutativity of rings $\omega(\hat{A})$ and $\omega(\check{B})$ inclusions take place:

[TABLE]

Hence $\bar{Z}=\mathop{\rm\bf Z}(\omega(\hat{A}))=\mathop{\rm\bf Z}(\omega(\check{B}))$. It means that rings $\omega(\hat{A})$ and $\omega(\check{B})$ are central $\bar{Z}$-algebras.

Because rings $\hat{A},\check{B}$ are primary then isomorphisms take place: $\omega(\hat{A})\cong\overline{\hat{A}}$, $\omega(\check{B})\cong\overline{\check{B}}$, and rings $\omega(\hat{A})$, $\omega(\check{B})$ are simple algebras.

Since modules ${}_{Z}\hat{A}$, ${}_{Z}\check{B}$ are finitely generated then $\bar{Z}$-algebras $\omega(\hat{A})$, $\omega(\check{B})$ are of finite dimension.

So $\omega(\hat{A}),\omega(\check{B})\in{\mathfrak{B}}(\bar{Z})$. According to Structure Theorem of Wedderburn–Artin there exist algebras with division (body) $\Delta,\nabla$ and naturals $r,s$ such that $\omega(\hat{A})=\Delta_{r,r}$, $\omega(\check{B})=\nabla_{s,s}$. Herewith $\mathop{\rm\bf Z}(\Delta)=\mathop{\rm\bf Z}(\nabla)=\bar{Z}$.

Because $\bar{C}=[\omega(\hat{A}),\omega(\check{B})]$ and rings $\omega(\hat{A})$, $\omega(\check{B})$ are pair-wise commutative then the socle ${\mathfrak{S}}({}_{C}M)$ is $(\omega(\hat{A}),\omega(\check{B})^{op})$-bimodule. Consequently according to Statement 3 there exists an epimorphism $\lambda:\omega(\hat{A})\mathop{\otimes}\limits_{\bar{Z}}\omega(\check{B})\to\bar{C}$. Herewith $\ker\lambda=l_{\omega(\hat{A})\mathop{\otimes}\limits_{\bar{Z}}\omega(\check{B})}({\mathfrak{S}}(M))$ is a left annihilator of module ${\mathfrak{S}}(M)$ in ring $\omega(\hat{A})\mathop{\otimes}\limits_{\bar{Z}}\omega(\check{B})$. Because according to [25, $\S 12.4$, Proposition b] $\omega(\hat{A})\mathop{\otimes}\limits_{\bar{Z}}\omega(\check{B})\cong(\Delta\mathop{\otimes}\limits_{\bar{Z}}\nabla)_{rs,rs}$ is a simple algebra and left module ${}_{\omega(\hat{A})\mathop{\otimes}\limits_{\bar{Z}}\omega(\check{B})}{\mathfrak{S}}({}_{C}M)$ is non-zero then $\ker\lambda=\hat{0}$. Thus

[TABLE]

Since $\bar{C}\cong{\bar{Z}}_{n,n}$ then Brauer classes of equivalence of $[\Delta\otimes\nabla]$ and $[\bar{Z}]$ coincide i.e. classes $[\Delta]$ and $[\nabla]$ are [25, $\S 12.5$, Proposition a] mutually inverse elements of group $\mathop{\rm\bf B}(\bar{Z})$ and this equalities take place: $[\Delta]^{-1}=[\Delta^{op}]=[\nabla]$. Thereby [25, $\S 12.5$, Proposition b(ii)] $\Delta^{op}\cong\nabla$.

Because of the (3.10) there exists an isomorphism $\mu:\omega(\hat{A})\to\bar{A}$. Since $\mathop{\rm\bf Z}(\omega(\hat{A}))=\bar{Z}$ then $\mathop{\rm\bf Z}(\bar{A})=\mu(\bar{Z})\cong\bar{Z}$. Consequently over an algebra $\bar{A}$ may be established the structure of $\bar{Z}$-module by have a put for all $\bar{a}\in\bar{A},\bar{z}\in\bar{Z}$ $\bar{z}\cdot\bar{a}=\mu\left(\bar{z}\mu^{-1}(\bar{a})\right)$. Then $\bar{A}$ is finite dimension central simple $\bar{Z}$-algebra because it inherits properties of $\bar{Z}$-algebra $\omega(\hat{A})$. Herewith basic algebras of $\bar{A}$ and $\omega(\hat{A})$ are isomorphic to algebra with division $\Delta$.

Analogously because of (3.11) there exists an isomorphism $\nu:\omega(\check{B})^{op}\to\bar{B}$. As above $\mathop{\rm\bf Z}(\omega(\check{B}))\cong\mathop{\rm\bf Z}(\omega(\check{B})^{op})\cong\bar{Z}$ what mean $\mathop{\rm\bf Z}(\bar{B})=\nu(\bar{Z})\cong\bar{Z}$. Structure of $\bar{Z}$-module over $\bar{B}$ is given by: for all $\bar{b}\in\bar{B},\bar{z}\in\bar{Z}$ let’s put $\bar{z}\cdot\bar{b}=\nu\left(\bar{z}\nu^{-1}(\bar{b})\right)$. Now $\bar{B}$ inherits properties of $\bar{Z}$-algebra $\omega(\check{B})^{op}$ and becomes because of that finite dimensional central simple $\bar{Z}$-algebra. Because basic algebras of $\bar{B}$ and $\omega(\check{B})^{op}$ are isomorphic to algebra with division $\nabla^{op}$ then we have only to note that since $\Delta\cong\nabla^{op}$ basic algebras of $\bar{Z}$-algebras $\bar{A}$ and $\bar{B}$ are isomorphic too.

I(d). Because of [25, $\S 12.4$, Proposition b(iv)] an isomorphism takes place:

[TABLE]

where $u=\mathop{\rm dim}_{\bar{Z}}\Delta$.

Hence according to (3.14) the equality takes place:

[TABLE]

Now let’s calculate dimension of the vector space ${\mathfrak{S}}(M)_{\bar{Z}}$ in other way.

Let $e_{i,j},i,j=\overline{1,r}$, $f_{u,v},u,v=\overline{1,s}$ be a full systems of matrix units of rings $\omega(\hat{A})$ and $\omega(\check{B})$ respectively. Since ${\mathfrak{S}}(M)$ is $(\omega(\hat{A}),\omega(\check{B})^{op})$-bimodule then two-sided Pierce decomposition of module ${\mathfrak{S}}(M)$ into direct sum of Abelian groups:

[TABLE]

Every summand $e_{i,i}{\mathfrak{S}}(M)f_{u,u}$, $i=\overline{1,r},u=\overline{1,s}$, is a $(e_{i,i}\omega(\hat{A})e_{i,i},f_{u,u}\omega(\check{B})^{op}f_{u,u})$-bimodule. All these bimodules are pair-wise isomorphic and because

[TABLE]

[TABLE]

the isomorphism takes place:

[TABLE]

where ${\mathfrak{S}}=e_{1,1}{\mathfrak{S}}(M)f_{1,1}$.

From relation (3.18) equalities follow:

[TABLE]

where $\mathop{\rm dim}_{\Delta}{\mathfrak{S}}$ and $\mathop{\rm dim}{\mathfrak{S}}_{\Delta}$ are dimensions of left ${}_{\Delta}{\mathfrak{S}}$ and right ${\mathfrak{S}}_{\Delta}$ vector spaces over the body $\Delta$.

Comparing (3.16) and (3.19) we find that

[TABLE]

Consequently

[TABLE]

is a quasi-Frobenius bimodule. Besides that from (3.21) follows an existence of isomorphism

[TABLE]

which implies an existence of isomorphism

[TABLE]

Since $\omega(\hat{A})\cong\bar{A}$, $\omega(\check{B})^{op}\cong({\bar{B}}^{op})^{op}={\bar{B}}$ then from existence of isomorphism (3.23) follows the existence of isomorphism

[TABLE]

Statement 10 shows that bimodule ${}_{\Delta_{r,r}}{\Delta_{r,s}}_{\Delta_{s,s}}$ is a quasi-Frobenius. Consequently because of (3.24) bimodule ${}_{\bar{A}}{\mathfrak{S}}({}_{C}M)_{\bar{B}}$ is also quasi-Frobenius.

So point I of this Theorem is proven completely.

II. Now we can prove point II of this Theorem. Let ${\mathfrak{S}}({}_{A}M)={\mathfrak{S}}(M_{B})={\mathfrak{S}}({}_{C}M)$. As we already have established bimodule ${}_{\bar{A}}{\mathfrak{S}}({}_{C}M)_{\bar{B}}$ is quasi-Frobenius. Hence because of the Statement 8 bimodule ${}_{A}M_{B}$ is also quasi-Frobenius.

Conversely if bimodule ${}_{A}M_{B}$ is quasi-Frobenius then according to [2, Теорема 1] ${\mathfrak{S}}({}_{A}M)={\mathfrak{S}}(M_{B})$. From other side

[TABLE]

[TABLE]

Since $J(\hat{A}),J(\check{B})\lhd J(C)$ then

[TABLE]

Consequently quasi-Frobenius bimodule ${}_{\bar{A}}{\mathfrak{S}}({}_{C}M)_{\bar{B}}$ is subbimodule of quasi-Frobenius bimodule ${}_{\bar{A}}{\mathfrak{S}}(M){}_{\bar{B}}$. This inclusion with necessity have to be an equality i.e.

[TABLE]

III. Since Brauer group $\mathop{\rm\bf B}(\bar{Z})$ of the field $\bar{Z}$ is trivial then $\Delta\cong\nabla\cong\bar{Z}$. Consequently ${}_{\Delta}{\mathfrak{S}}_{\Delta}\cong{}_{\bar{Z}}{\bar{Z}}_{\bar{Z}}$. Because of above and (3.24) an isomorphism takes place:

[TABLE]

Besides that from (3.16) the equality follows: $n=r\cdot s$.

Let’s show now that $A\cong Z_{r,r}$, $B\cong Z_{s,s}$. Indeed rings $A$ and $B$ contain as subrings $\phi(Z_{r,r})\cong Z_{r,r}$ and $\psi(Z_{s,s})\cong Z_{s,s}$ respectively. Let’s denote by $\hat{\phi}(Z_{s,s})$ and $\check{\psi}(Z_{s,s})$ images of those subrings under converse mapping into $C$. Let’s denote by $\tilde{C}$ ring generated by $\hat{\phi}(Z_{r,r})$ and $\check{\psi}(Z_{s,s})$ in $C$. According to Statement 3 ring $\tilde{C}$ is an epimorphic image of ring

[TABLE]

i.e.

[TABLE]

Since $C\cong Z_{n,n}$ and $n=rs$ then an isomorphism takes place:

[TABLE]

which implies isomorphisms:

[TABLE]

Because left module ${}_{C}M$ is faithful and according to (3.27) left module

[TABLE]

is also faithful. In view of isomorphism (3.26) and isomorphisms $\hat{\phi}(Z_{r,r})\cong Z_{r,r}$, $\check{\psi}(Z_{s,s})\cong Z_{s,s}$ we conclude that $\tilde{C}\cong C$. Since $\tilde{C}=[\hat{\phi}(Z_{r,r}),\check{\psi}(Z_{s,s})]<[\hat{A},\check{B}]=C$ then ring $\tilde{C}$ is embedded identically into ring $C$ and previous isomorphism is only an equality. Consequently $\hat{\phi}(Z_{r,r})=\hat{A}$, $\check{\psi}(Z_{s,s})=\check{B}$, where from we get isomorphisms of point III(a) of this Theorem.

Let’s now trace a chain of isomorphisms:

[TABLE]

From other side

[TABLE]

Here we have used a generalization of Proposition from $\S 10.1$ of [25]. So we have established an isomorphism of point III(b) of this Theorem.

Bimodule ${}_{Z_{r,r}}{Q_{r,s}}_{Z_{s,s}}$ is quasi-Frobenius because of the Statement 10. Bimodule ${}_{A}M_{B}$ is quasi-Frobenius because of the isomorphism of point III(b) of this Theorem. ∎

4 Proof of the Theorem 6

For convenience let’s remember that the ring $S$ is called * Galois-Eisenstein-Ore ring * (or * $GEO$-ring *) if it is finite completely primary ring of principal ideals i.e. the ring $S$ contains unique (one-side) maximal ideal ${\mathfrak{p}}(S)$ and every one-side ideal of ring $S$ is principal. Commutative $GEO$-ring is called Galois-Eisenstein ring or $GE$-ring [18].

In arbitrary $GEO$-ring $S$ of characteristic $p^{d}$ contains

• Galois subring * $R=GR(p^{d},r)$ [13, 26], $q=p^{r}$. All such rings are conjugated in $S$. Ring $R$ is called *coefficients ring * of $S$.

And also let’s repeat conditions of Theorem 6:

Theorem 12**.**

(a) For arbitrary $GEO$-ring $S$ bimodule ${}_{S}S_{S}$ is a quasi-Frobenius.

(b) Let $S$ be a $GEO$-ring with a coefficients ring $R=GR(p^{d},r)$ and in addition $d>1$ and $(r,p)=1$. Then bimodule of translations ${}_{C}S_{Z}$ of quasi-Frobenius $S$-bimodule $S$ is a quasi-Frobenius bimodule if and only if $S$ is a $GE$-ring.

Let’s denote by $n(S)$ the nilpotency index of ideal ${\mathfrak{p}}(S)$. The field $S/{\mathfrak{p}}(S)$ we denote by $\bar{S}$ and an image of $s\in S$ in $\bar{S}$ by $\bar{s}$. Let $\bar{S}=GF(q)$.

For every finite completely primary ring $S$ following statements are equivalent [18, Theorem 1.1, points I, II, VI, VII, VIII respectively ]:

(a) $S$ is a $GEO$-ring;

(b) $|S|=q^{n(S)}$;

(c) Every one-side ideal of $S$ is a degree of ${\mathfrak{p}}(S)$;

(d) For every $\pi_{t}\in{\mathfrak{p}}(S)^{t}\setminus{\mathfrak{p}}(S)^{t+1}$, $t=\overline{0,n(S)-1}$, this equalities take place: ${\mathfrak{p}}(S)^{t}=\pi_{t}S=S\pi_{t}$ (if $t=0$ then we denote ${\mathfrak{p}}(S)^{0}=S$).

*(e) If $\gamma:\bar{S}\to S$ is a mapping with property *

[TABLE]

*and $\pi_{t}\in{\mathfrak{p}}(S)^{t}\setminus{\mathfrak{p}}(S)^{t+1}$, $t=\overline{0,n(S)-1}$, then an arbitrary element $s\in S$ is uniquely represented in the form: $s=\sum\limits_{t=0}^{n(S)-1}s_{t}\pi_{t}$, where $s_{t}\in\gamma(\bar{S})$, $t=\overline{0,n(S)-1}$, and also in the form: $s=\sum\limits_{t=0}^{n(S)-1}\pi_{t}s_{t}^{\prime}$, where $s_{t}^{\prime}\in\gamma(\bar{S})$, $t=\overline{0,n(S)-1}$. *

Let’s view $S$ as left unitary $R$-module. System of elements $s_{1},\ldots,s_{t}\in{}_{R}S$ is called [18, $\S 2$] *free * if there no exists a non-trivial linear relation between these elements over $R$ and is called irreducible if no one of these elements is a linear combination over $R$ of others. Irreducible generating system of ${}_{R}S$ is called * basis * of ${}_{R}S$. By *dimension * $\mathop{\rm dim}{}_{R}S$ is called a number of elements in basis of ${}_{R}S$ which is equal to dimension of vector space $S/{\mathfrak{p}}(R)S$ over field $\bar{R}$.

Let $T({}_{R}S)=\{s\in S\ |\ {\mathfrak{p}}(R)^{n(R)-1}s=0\}$ be a set of elements of ${}_{R}S$ annihilated by some non-zero element from $R$. Then [18, Proposition 2.1] the number of elements in maximal free subsystem of ${}_{R}S$ satisfies to equality: $\mathop{\rm rang}{}_{R}S=\mathop{\rm dim}_{\bar{R}}(S/T({}_{R}S))$.

It is known [18, Corollary 2.3] that the equality takes place: $\mathop{\rm dim}{}_{R}S=\mathop{\rm dim}S_{R}$, $\mathop{\rm rang}{}_{R}S=\mathop{\rm rang}S_{R}$, and besides that an arbitrary basis of ${}_{R}S$ is also a basis of $S_{R}$ and conversely. Because of that we can speak simply about rank, dimension and basis of $S$ over $R$.

For some natural $e=e(S|R)$ [18, Theorem 1.1] these equalities take place:

[TABLE]

and

[TABLE]

Besides that $n(S)=(n(R)-1)e+\rho$ where $\rho=\rho(S|R)$ satisfies to inequalities: $1\leq\rho\leq e$. Herewith [18, Proposition 5.1] * $S$ is a $R$-bimodule of the rank $\rho=\rho(S|R)$ and dimension $e=e(S|R)$ whereas $T(S)={\mathfrak{p}}(S)^{\rho}$. *

Let $\sigma\in\mathop{\rm Aut}(R)$. * By the ring of Ore polynomials * $R[x,\sigma]$ is called ring of polynomials with usual addition and multiplication established by the rule: $x\cdot r=r^{\sigma}\cdot x,r\in R$. If $t=t(\sigma)$ is an order of $\sigma$, then center $\mathop{\rm\bf Z}(R[x,\sigma])$ of ring $R[x,\sigma]$ contains of and only of polynomials of the form $c_{0}+c_{t}x^{t}+\cdots+c_{tm}x^{tm}$ where $c_{0},\ldots,c_{tm}\in R_{\sigma}=\{r\in R\ |\ r^{\sigma}=r\}$.

Polynomial $x^{l}+c_{l-1}x^{l-1}+\cdots+c_{0}\in R[x,\sigma]$ is called * Eisenstein polynomial * if $c_{i}\in{\mathfrak{p}}(R)$ for $i=\overline{0,l-1}$ and $c_{0}\not\in{\mathfrak{p}}(R)^{2}$ if $n(R)>1$. Eisenstein polynomial of the form

[TABLE]

is called * special * if either

[TABLE]

either for some $a<m$

[TABLE]

It is known [18, Theorem 5.2 II] that * if $R$ is a ring of coefficients of $GEO$-ring $S$, $n(R)>1$, $e=e(S|R)$, $\rho=\rho(S|R)$, then there exist an automorphism $\sigma\in\mathop{\rm Aut}(R)$ such that $t=t(\sigma)$ divides $e$, $e=t\cdot m$, and special Eisenstein polynomial of the form (4.2) with the property:*

[TABLE]

where $I=c(x)R[x,\sigma]+x^{\rho}{\mathfrak{p}}(R)^{n(R)-1}[x,\sigma]$. Herewith (4.4) is fulfilled only if $\rho=ta+1$.

Automorphism $\sigma$ in (4.5) is uniquely determined by the ring $S$ and do not depends on the choice of coefficient ring $R$ [18, Proposition 5.5].

Let’s remember that $GEO$-ring $S$ is a * Galois–Eisenstein ring ($GE$-ring )* iff $t=t(\sigma)=1$ i.e. Galois–Eisenstein ring is nothing but commutative Galois–Eisenstein–Ore ring [17].

Besides that [18, Proposition 5.7] *if $\pi$ is a $\sigma$-element from ${\mathfrak{p}}(S)\setminus{\mathfrak{p}}(S)^{2}$ ( i.e. for every $r\in R$ this equality satisfies: $\pi r=r^{\sigma}\pi$) then: *

I) The centralizer $C(R)$ of ring $R$ in $S$ is a $GE$-ring ([17])

[TABLE]

whereas $e(C(R)|R)=m$, $\rho(C(R)|R)=\rho^{\prime}$, $\rho^{\prime}=\left[\frac{\rho}{t}\right]$, and

[TABLE]

whereas $c(x)=z(x^{t})$.

• II) The center $\mathop{\rm\bf Z}(S)$ is equal to*

[TABLE]

whereas $\rho\not\equiv 1(\mathop{\rm mod}t)$ and is equal to

[TABLE]

*in other case. *

It is known [7, $\S 13$] that arbitrary left-side and right-side together Artinian ring $T$ with identity is * quasi-Frobenius * if and only if $T$-bimodule $T$ is a quasi-Frobenius.

$\mathop{\rm\bf Proof\ of\ the\ Theorem\ \ref{TH4+1}}$.

(a) Let’s use the Statement 8. These equalities take place:

[TABLE]

whereas $\pi\in{\mathfrak{p}}(S)\setminus{\mathfrak{p}}(S)^{2}$. Analogously

[TABLE]

This way ${\mathfrak{S}}({}_{S}S)={\mathfrak{S}}(S_{S})={\mathfrak{S}}(S)$. Besides that $\mathop{\rm dim}{\mathfrak{S}}(S)_{\bar{S}}=\mathop{\rm dim}_{\bar{S}}{\mathfrak{S}}(S)=1$. Hence according to point (4) of Theorem 7 bimodule ${}_{\bar{S}}{\mathfrak{S}}(S)_{\bar{S}}$ is quasi-Frobenius. Thus bimodule ${}_{S}S_{S}$ and ring $S$ is also quasi-Frobenius.

(b) Let’s note that $n(R)=d$. Because $d>1$ the ring $S$ has a form:

[TABLE]

where $\rho=\rho(S|R)$, $e=e(S|R)$, $t=t(\sigma)$, $\sigma\in\mathop{\rm Aut}(R)$, $I=c(x)R[x,\sigma]+x^{\rho}{\mathfrak{p}}(R)^{n(R)-1}[x,\sigma]$, $c(x)=x^{tm}+c_{t(m-1)}x^{t(m-1)}+\cdots+c_{0}$ is a special Eisenstein polynomial, whereas condition

[TABLE]

is satisfied only if $\rho=ta+1$.

Besides that by $r=\log_{p}q$ we have $t|r$. Since under condition $(r,p)=1$ we have then $(t,p)=1$.

Let $\cal R$ be a ring of translations of $R$-bimodule $S$, $C$ be a ring of translations of $S$-bimodule $S$.

Let’s prove that ring $\cal R$ is isomorphic to foreign direct sum of $\min\{t,\rho\}$ copies of ring $R=GR(p^{d},r)$ and $\max\{0,t-\rho\}$ copies of ring $\tilde{R}=R/{\mathfrak{p}}(R)^{n(R)-1}=GR(p^{d-1},r)$.

Let’s denote by $\epsilon_{l}\in{\cal R}$, $l=\overline{0,t-1}$, endomorphisms of $R$-bimodule $S$ of the form:

[TABLE]

operating according to the rule:

[TABLE]

where $\delta_{\lambda,\mu}$ is a Kronecker symbol.

Since equalities take place: $S=\sum_{j=0}^{e-1}R\pi^{j}$ and $c(\pi)=0$ where $\pi$ is a $\sigma$-element from ${\mathfrak{p}}(S)\setminus{\mathfrak{p}}(S)^{2}$ we have:

[TABLE]

Hence

[TABLE]

It is evident that ring of translations $\cal R$ of $R$-bimodule $S$ may be represented in the form: ${\cal R}=\sum\limits_{r_{1},r_{2}\in R}\hat{r}_{1}\check{r}_{2}$. Let’s operate the structure of this set. To do this let’s describe previously the ring $\check{R}$.

Let $r\in R$ and $s=\sum_{l=\overline{0,e-1}}r_{l}\pi^{l}\in S$. Then

[TABLE]

since $r^{\sigma^{l}}=r^{\sigma^{j}}$ if and only if $l\equiv j(\mathop{\rm mod}t)$.

Define mapping $\phi_{l}:R\to\mathop{\rm End}(S)$, $l=\overline{0,t-1}$, by the rule: for every $r\in R$ put

[TABLE]

i.e.

[TABLE]

Thus

[TABLE]

Let’s show that the family of set $\phi_{l}(R)$, $l=\overline{0,t-1}$ are pair-wise orthogonal subrings of the ring $\mathop{\rm End}(S)$. To do this, it suffices to prove that in fact there are elements $\epsilon_{l}\in{\cal R}$, $l=\overline{0,t-1}$, with pointed in equality (4.12) property which it is convenient to represent in the form

[TABLE]

It is evident that an arbitrary mapping $\beta\in{\cal R}$ may be represented in the following form:

[TABLE]

We pose the mapping $\beta$ into compliance a vector $\vec{\beta}=(\beta_{0},\ldots,\beta_{t-1})$, $\beta_{l}\in R,l=\overline{0,t-1}$, and determine the action of this vector on the set $S$ in this way:

[TABLE]

Let $\beta,\gamma\in{\cal R}$, $r\in R$. Then mapping $\zeta=\hat{r}\cdot\beta\in{\cal R}$ is represented by vector $\vec{\zeta}=r\cdot\vec{\beta}$, and mapping $\xi=\beta+\hat{r}\cdot\gamma\in{\cal R}$ is represented by vector $\vec{\xi}=\vec{\beta}+r\cdot\vec{\gamma}$. Converse is also true.

Let’s remember that for $r\in R$

[TABLE]

Let’s denote this vector as $\vec{\beta}(r)$ and consider the matrix

[TABLE]

Since $(t,p)=1$ over field $\bar{R}=GF(q)$ there are exactly $t$ different roots from unit of degree $t$ namely: $\aleph_{1},\ldots,\aleph_{t}$. Let $f_{r}(x)=\bar{r}+\overline{r^{\sigma}}\cdot x+\cdots+\overline{r^{\sigma^{t-1}}}\cdot x^{t-1}\in\bar{R}[x]$. Then there is an equality:

[TABLE]

When $r\in R^{*}\setminus R_{\sigma}$ the right-hand side of previous equality differs from zero. Since with $r\in R^{*}\setminus R_{\sigma}$ and $(t,p)=1$ the matrix $C(r)$ is invertible. Hence by equivalent transformations of rows matrix $C(r)$ is possible to lead to the identity matrix $E_{t\times t}$. It remains to remark that for every $l\in\overline{0,t-1}$ row $\vec{E}_{l+1}$ is a vector of desired mapping $\epsilon_{l}\in{\cal R}$.

It is easy to see that there are equalities:

[TABLE]

from which it follows that $\phi_{l}(R)$, $l=\overline{0,t-1}$, are pair-wise orthogonal subrings of ring $\cal R$.

Besides that now the equality is evident:

[TABLE]

Let’s describe now rings $\phi_{l}(R)$, $l=\overline{0,t-1}$. To do this let’s note that for arbitrary $r\in R$ $\phi_{l}(r)=\check{0}$ if and only if $r^{\sigma^{l}}\cdot\pi^{l}=0$, i.e. $r^{\sigma^{l}}\pi^{l}\in{\mathfrak{p}}(S)^{n(S)}$, whereas $n(S)=e(n(R)-1)+\rho$, $0\leq\rho\leq e-1$. Let $r\in{\mathfrak{p}}(R)^{\kappa}$. Then $r^{\sigma^{l}}\pi^{l}\in{\mathfrak{p}}(S)^{e\kappa+l}$ and equality $r^{\sigma^{l}}\pi^{l}=0$ is achieved if and only if $e\kappa+l\geq e(n(R)-1)+\rho$, i.e. $l\in\overline{\rho,e-1}$, $\kappa=n(R)-1$. Let’s note now that $t\leq e$. Hence these isomorphisms take place:

[TABLE]

[TABLE]

Let’s describe now the ring $\bar{C}=C/J(C)$. Let’s note that $C={\cal R}[\hat{\pi},\check{\pi}]$. Since $\hat{\pi}$ and $\check{\pi}$ are nil-potent elements of ring $C$ then $\hat{\pi},\check{\pi}\in J(C)$. Consequently $\bar{C}=C/J(C)\cong\mathop{\sum\oplus}\limits_{l=\overline{0,t-1}}\overline{\phi_{l}(R)}$ is isomorphic to foreign direct sum of $t$ copies of the field $\bar{R}=GF(q)$.

Immediately from the definitions it follows that for arbitrary $GEO$-ring $S$ common center $Z$ of rings $\hat{S}$ and $\check{S}$ is equal to $\mathop{\rm\bf Z}(S)$.

Let’s suppose now that bimodule ${}_{C}S_{Z}$ is quasi-Frobenius. Then since $Z$ is local Artinian ring we can apply results of Theorem 5. In particular according to point I(a) of Theorem 5 the ring $C$ is isomorphic to $Z_{n,n}$ whereas $n=\mathop{\rm dim}\nolimits_{\bar{Z}}{\mathfrak{S}}(S)$. Let’s remark that $\bar{Z}=\overline{R_{\sigma}}$ and $\mathop{\rm dim}_{\bar{Z}}{\mathfrak{S}}(S)=t=\mathop{\rm ord}\sigma$.

Thus ring $\bar{C}=C/J(C)\cong\mathop{\sum\oplus}\limits_{l=\overline{0,t-1}}\overline{\phi_{l}(R)}$ is isomorphic to foreign direct sum of $t$ copies of the field $\bar{R}=GF(q)$ and has to be isomorphic to ring of $t\times t$-matrices over ring ${R_{\sigma}}$.

But the ring $\bar{C}$ is isomorphic to matrix ring only if $t=1$ i.e. if $S$ is a $GE$-ring.

Converse implication is evident because firstly an arbitrary $GE$-ring is quasi-Frobenius and secondly for arbitrary commutative ring $S$ ring of translations of $S$-bimodule $S$ and common center of $\hat{S}$ and $\check{S}$ relatively to $S$ coincide with $S$. $\Box$