Automorphism-invariant non-singular rings and modules

TL;DR

This paper characterizes automorphism-invariant non-singular rings and modules, establishing their structure and properties through equivalences involving regularity, injectivity, and Goldie conditions.

Contribution

It provides new characterizations and structural descriptions of automorphism-invariant non-singular rings and modules, linking them to regularity and injectivity properties.

Findings

A ring is automorphism-invariant non-singular iff it decomposes into specific regular and strongly regular components.

Conditions under which direct sums of automorphism-invariant modules are injective or automorphism-invariant are established.

The structure of rings with certain Goldie radical properties is characterized through module invariance and semiprimeness.

Abstract

$\textbf{Theorem 1.2.}$ For a ring $A$, the following conditions are equivalent. $\textbf{1)}$ $A$ is a right automorphism-invariant right non-singular ring. $\textbf{2)}$ $A$ is a right automorphism-invariant regular ring. $\textbf{3)}$ $A=S\times T$, where $S$ is a right injective regular ring and $T$ is a strongly regular ring which contains all invertible elements of its maximal right ring of quotients. $\textbf{Theorem 1.5.}$ For a ring $A$ with right Goldie radical $G(A_A)$, the following conditions are equivalent. $\textbf{1)}$ $A/G(A_A)$ is a semiprime right Goldie ring. $\textbf{2)}$ Any direct sum of automorphism-invariant non-singular right $A$-modules is an automorphism-invariant module. $\textbf{3)}$ Any direct sum of automorphism-invariant non-singular right $A$-modules is an injective module.

Full text

Automorphism-Invariant Non-Singular Rings and Modules

A.A. Tuganbaev111National Research University ”MPEI”, Lomonosov Moscow State University,

e-mail: tuganbaev@gmail.com.

Abstract. A ring $A$ is a right automorphism-invariant right non-singular ring if and only if $A=S\times T$, where $S$ a right injective regular ring and $T$ is a strongly regular ring which contains all invertible elements of its maximal right ring of quotients. Over a ring $A$, each direct sum of automorphism-invariant non-singular right modules is an automorphism-invariant module if and only if the factor ring of the ring $A$ with respect to its right Goldie radical is a semiprime right Goldie ring.

The study is supported by Russian Scientific Foundation (project 16-11-10013).

Key words: automorphism-invariant ring, automorphism-invariant module, injective module, quasi-injective module

1. Introduction and preliminaries

All rings are assumed to be associative and with zero identity element; all modules are unitary. A module $M$ is said to be automorphism-invariant if $M$ is invariant under any automorphism of its injective hull. In [2] Dickson and Fuller studied automorphism-invariant modules, when the underlying ring is a finite-dimensional algebra over a field with more than two elements. In [3, Theorem 16] Er, Singh and Srivastava proved that a module $M$ is automorphism-invariant if and only if $M$ is a pseudo-injective module, i.e., for any submodule $X$ of $M$, every monomorphism $X\to M$ can be extended to an endomorphism of the module $M$. Pseudo-injective modules were studied in several papers; e.g., see [8], [15], [3]. Automorphism-invariant modules were studied in several papers; e.g., see [1], [3], [5], [11], [13], [16], [17], [18], [19], [20].

A ring $A$ is said to be regular if every its principal right (left) ideal is generated by an idempotent. A ring $A$ is said to be strongly regular if every its principal right (left) ideal is generated by a central idempotent. A module $X$ is said to be injective relative to the module $Y$ or $Y$-injective if for any submodule $Y_{1}$ of $Y$, every homomorphism $Y_{1}\to X$ can be extended to a homomorphism $Y\to X$. A module is said to be injective if it is injective with respect to any module. A module is said to be square-free if it does not contain a direct sum of two non-zero isomorphic submodules. A submodule $Y$ of the module $X$ is said to be essential in $X$ if $Y\cap Z\neq 0$ for any non-zero submodule $Z$ of $X$. A submodule $Y$ of the module $X$ is said to be closed in $X$ if $Y=Y^{\prime}$ for every submodule $Y^{\prime}$ of $X$ which is an essential extension of the module $Y$. We denote by $\operatorname{Sing}X$ the singular submodule of the right $A$-module $X$, i.e., $\operatorname{Sing}X$ is a fully invariant submodule of $X$ which consists of all elements $x\in X$ such that $r(x)$ is an essential right ideal of the ring $A$. A module $X$ is said to be non-singular if $\operatorname{Sing}X=0$.

Remark 1.1. In [3, Theorem 7, Theorem 8, Example 9] Er, Singh and Srivastava proved the following results.

(1) If $A$ is a right non-singular, right automorphism-invariant ring, then $A=S\times T$, where the ring $S$ is right injective, the module $T_{T}$ is square-free, any sum of closed right ideals of $T$ is a two-sided ideal which is an automorphism-invariant right $T$-module, and for any prime ideal $P$ of $T$ which is not essential in $T_{T}$, the factor ring $T/P$ is a division ring.

(2) If $A$ is a right non-singular, right automorphism-invariant prime ring, then the ring $A$ is right injective.

(3) Let $F$ be the field of order 2, $S$ the direct product of a countable set of copies of $F$, and $A=\{(f_{n})_{n=1}^{\infty}\in S$: almost all $f_{n}$ are equal to some $a\in F\}$. Then $A$ is a commutative automorphism-invariant regular ring, but it is not an injective $A$-module.

In connection to Remark 1.1, we will prove Theorem 1.2 which is the first main result of the given paper.

Theorem 1.2. For a ring $A$, the following conditions are equivalent.

1. 1)

$A$* is a right automorphism-invariant right non-singular ring.* 2. 2)

$A$* is a right automorphism-invariant regular ring.* 3. 3)

$A=S\times T$, where $S$ is a right injective regular ring and $T$ is a strongly regular ring which contains all invertible elements of its maximal right ring of quotients.

Remark 1.3. A module $X$ is said to be quasi-injective if $X$ is injective relative to $X$, i.e., for any submodule $X_{1}$ of $X$, every homomorphism $X_{1}\to X$ can be extended to an endomorphism of the module $X$. Every quasi-injective module is an automorphism-invariant module, since the module $X$ is quasi-injective if and only if $X$ is invariant under any endomorphism of its injective hull; e.g., see [10, Theorem 6.74]. Every finite cyclic group is a quasi-injective non-injective module over the ring $\mathbb{Z}$ of integers.

For a module $X$, we denote by $G(X)$ the intersection of all submodules $Y$ of the module $X$ such that the factor module $X/Y$ is non-singular. The submodule $G(X)$ is a fully invariant submodule of $X$; it is called the Goldie radical of the module $X$.

Remark 1.4. In [9, Theorem 3.8] Kutami and Oshiro proved that any direct sum of non-singular quasi-injective right modules over the ring $A$ is quasi-injective if and only if $A/G(A_{A})$ is a semiprime right Goldie ring.

In connection to Remark 1.4, we will prove Theorem 1.5 which is the second main result of the given paper.

Theorem 1.5. For a ring $A$ with right Goldie radical $G(A_{A})$, the following conditions are equivalent.

1. 1)

$A/G(A_{A})$* is a semiprime right Goldie ring.* 2. 2)

Any direct sum of automorphism-invariant non-singular right $A$-modules is an automorphism-invariant module. 3. 3)

Any direct sum of automorphism-invariant non-singular right $A$-modules is an injective module.

The proof of Theorem 1.2 and Theorem 1.5 is decomposed into a series of assertions, some of which are of independent interest.

We give some necessary definitions. A module $X$ is said to be singular if $X=\operatorname{Sing}X$. A module $X$ is a Goldie-radical module if $X=G(X)$. The relation $G(X)=0$ is equivalent to the property that the module $M$ is non-singular. In the paper, we use well-known properties of $\operatorname{Sing}X$, $G(X)$, non-singular modules and maximal right rings of quotients; e.g., see [4, Chapter 2], [10, Section 7] and [12, Section 3.3]. A module $Q$ is called an injective hull of the module $M$ if $Q$ is an injective module and $M$ is an essential submodule of the module $Q$. A module $M$ is called a CS module if every its closed submodule is a a direct summand of the module $M$. A module $M$ is said to be uniform if the intersection of any two non-zero submodules of the module $M$ is not equal to zero. A module $M$ is said to be finite-dimensional if $M$ does not contain an infinite direct sum of non-zero submodules.

A ring $A$ is called a right Goldie ring if $A$ is a right finite-dimensional ring with the maximum condition on right annihilators. A ring $A$ is said to be reduced if $A$ does not have non-zero nilpotent elements. A ring without non-zero nilpotent ideals is said to be semiprime ring. A ring $A$ is said to be right strongly semiprime [6] if any its ideal, which is an essential right ideal, contains a finite subset with zero right annihilator. A ring is said to be right strongly prime [7] if every its non-zero ideal contains a finite subset with zero right annihilator.

Remark 1.6. Every right strongly semiprime ring is a right non-singular semiprime ring [6]. It is clear that every right strongly prime ring is right strongly semiprime. The direct product of two finite fields is a finite commutative strongly semiprime ring which is not strongly prime. The direct product of a countable number of fields is an example of a commutative semiprime non-singular ring which is not strongly semiprime. All finite direct products of rings without zero-divisors and all finite direct products of simple rings are right and left strongly semiprime rings.

Remark 1.7. If $A$ is a semiprime right Goldie ring, then it is well known222For example, see [12, Theorem 3.2.14]. that every essential right ideal of the ring $A$ contains a non-zero-divisor. Therefore, all semiprime right Goldie rings are right strongly semiprime. In particular, all right Noetherian semiprime rings are right strongly semiprime.

2. Automorphism-Invariant Nonsingular Rings

Lemma 2.1 [12, Section 3.3.]. Let $A$ be a right non-singular ring with maximal right ring of quotients $Q$. Then $Q$ is an injective right regular ring and $Q$ can be naturally identified with the ring $\operatorname{End}Q_{A}$ and $Q_{A}$ is an injective hull of the module $A_{A}$.

Lemma 2.2. If $A$ is a right non-singular ring with maximal right ring of quotients $Q$, then $A$ is a right automorphism-invariant ring if and only if $A$ contains all invertible elements of the ring $Q$.

Lemma 2.2 follows from Lemma 2.1.

Lemma 2.3 [14, Chapter 12, 5.1–5.4.]. Let $A$ be a reduced ring with maximal right ring of quotients $Q$. Then ring $A$ is right and left non-singular. If each of the closed right ideals of the ring $A$ is an ideal, then $A$ is a reduced ring and $Q$ is a right and left injective strongly regular ring.

Lemma 2.4. Let $A$ be a right non-singular ring in which all closed right ideals are ideals. Then the ring $A$ is reduced.

Proof. Let $a$ be an element of $A$ with $a^{2}=0$. There exists a closed right ideal $B$ of $A$ such that $B\cap aA=0$ and $B+aA$ is an essential right ideal of $A$. By assumption, the closed right ideal $B$ is an ideal. Therefore, $aB=0$, since $aB$ is contained in the intersection of $aA$ and $B$. Then $a(B+aA)=0$ and $B+aA$ is an essential right ideal. Since $A$ is right non-singular, $a=0$. $\square$

Lemma 2.5. If $A$ is a right automorphism-invariant right non-singular ring, then $A=S\times T$, where $S$ is a right injective regular ring and $T$ is a strongly regular ring which contains all invertible elements of its maximal right ring of quotients.

Proof. By Remark 1.1(1) and Lemma 2.1, $A=S\times T$, where $S$ is a right injective regular ring, $T$ is a right automorphism-invariant right non-singular ring, and any closed right ideal of $T$ is an ideal. By Lemma 2.4, $T$ is a reduced ring. Let $Q$ be the maximal right ring of quotients of the ring $T$. By Lemma 2.3, $T$ is a reduced ring and $Q$ is a right and left injective strongly regular ring. To prove that $T$ is a strongly regular ring, it is sufficient to prove that an arbitrary element $t$ of the ring $T$ is the product of a central idempotent and an invertible element. Since $t$ is an element of the strongly regular ring $Q$, we have that $t=eu$, where $e$ is a central idempotent of the ring $Q$ and $u$ is an invertible element of the ring $Q$. By Lemma 2.2, $T$ contains all invertible elements of the ring $Q$. Therefore, $u\in T$. Then $e=tu^{-1}\in T$ and every element of the ring $T$ is the product of a central idempotent and an invertible element. $\square$

Remark 2.6. The completion of the proof of Theorem 1.2. In Theorem 1.2, the implication 1) $\Rightarrow$ 3) follows from Lemma 2.5, the implication 3) $\Rightarrow$ 2) follows from the property that the direct product of regular rings $S$ and $T$ is a regular ring, and the implication 2) $\Rightarrow$ 1) follows from the property that every regular ring is right and left non-singular.

Corollary 2.7. If $A$ is a right automorphism-invariant right non-singular indecomposable ring, then $A$ is a right injective ring; see Remark 1.1(2).

Corollary 2.7 follows from Theorem 1.2 and the property that every strongly regular indecomposable ring is a division ring and, consequently, a right injective ring.

Corollary 2.8. Let $A$ be a right automorphism-invariant right non-singular ring which does not contain an infinite set of non-zero central orthogonal idempotents. Then $A$ is a right injective ring.

Corollary 2.8 follows from Corollary 2.7 and the property that every ring, which does not contain an infinite set of non-zero central orthogonal idempotents, is a finite direct product of indecomposable rings.

3. Automorphism-Invariant Non-singular Modules

Lemma 3.1. Let $A$ be a ring and $X$ a right $A$-module which is not an essential extension of a singular module. Then there exists a non-zero right ideal $B$ of the ring $A$ such that the module $B_{A}$ is isomorphic to a submodule of the module $X$.

Proof. Since the module $X$ is not an essential extension of a singular module, there exists an element $x\in X$ such that $xA$ is a non-zero non-singular module. Since $xA\cong A_{A}/r(x)$ and the module $xA$ is non-singular, the right ideal $r(x)$ is not an essential. Therefore, there exists a non-zero right ideal $B$ with $B\cap r(x)=0$. In addition, there exists an epimorphism $f\colon A_{A}\to xA$ with kernel $r(x)$. Since $B\cap\operatorname{Ker}f=0$, we have that $f$ induces the monomorphism $g\colon B\to xA$. Therefore, $xA$ contains the non-zero submodule $g(B)$ which is isomorphic to the module $B_{A}$.  $\square$

Lemma 3.2. Let $A$ be a ring, $G=G(A_{A})$ the right Goldie radical of the ring $A$, $h\colon A\to A/G$ the natural ring epimorphism and $X$ a non-singular non-zero right $A$-module.

1. 1)

If $B$ is a essential right ideal of the ring $A$, then $h(B)$ is an essential right ideal of the ring $h(A)$. 2. 2)

If $B$ is a right ideal of the ring $A$ such that $G\subseteq B$ and $h(B)$ is an essential right ideal of the ring $h(A)$, then $B$ is an essential right ideal of the ring $A$. 3. 3)

For any right $A$-module $M$, the module $MG$ is contained in the Goldie radical of $M$. 4. 4)

$XG=0$* and the natural $h(A)$-module $X$ is non-singular. In addition, if $Y$ is an arbitrary non-singular right $A$-module, then $YG=0$ and the $h(A)$-module homomorphisms $Y\to X$ coincide with the $A$-module homomorphisms $Y\to X$. Therefore, $X$ is an $Y$-injective $A$-module if and only if $X$ is an $Y$-injective $h(A)$-module. The essential submodules of the $h(A)$-module $X$ coincide with the essential submodules of the $A$-module $X$.* 5. 5)

$X$* is an injective $h(A)$-module if and only if $X$ is an injective $A$-module.* 6. 6)

$X_{h(A)}$* is a uniform module $($resp., an essential extension of a direct sum of uniform modules$)$ if and only if $X_{A}$ is a uniform module $($resp., an essential extension of a direct sum of uniform modules$)$.* 7. 7)

$X_{A}$* is an essential extension of a direct sum of modules each of them is isomorphic to some non-zero right ideal of the ring $A$.* 8. 8)

If the ring $A$ is right finite-dimensional, then $X_{A}$ is an essential extension of a direct sum of modules each of them is isomorphic to some non-zero uniform right ideal of the ring $A$. 9. 9)

If the ring $h(A)$ is right finite-dimensional, then $X_{h(A)}$ is an essential extension of a direct sum of modules each of them is isomorphic to some non-zero uniform right ideal of the ring $h(A)$.

Proof. 1. Let us assume that $h(B)$ is not an essential right ideal of the ring $h(A)$. Then there exists a right ideal $C$ of the ring $A$ such that $C$ properly contains $G$ and $h(B)\cap h(C)=h(0)$. Since $h(B)\cap h(C)=h(0)$, we have that $B\cap C\subseteq G$. Since $C$ properly contains the closed right ideal $G$, we have that $C_{A}$ contains a non-zero submodule $D$ with $D\cap G=0$. Since $B$ is an essential right ideal, $B\cap D\neq 0$ and $(B\cap D)\cap G=0$. Then $h(0)\neq h(B\cap D)\subseteq h(B)\cap h(C)=h(0)$. This is a contradiction.

2. Let us assume that $B$ is not an essential right ideal of the ring $A$. Then $B\cap C=0$ for some non-zero right ideal $C$ of the ring $A$ and $G\cap C\subseteq B\cap C=0$. Therefore, $h(C)\neq h(0)$. Since $h(B)$ is an essential right ideal of the ring $h(A)$, we have that $h(B)\cap h(C)\neq h(0)$. Let $h(0)\neq h(b)=h(c)\in h(B)\cap h(C)$, where $b\in B$ and $c\in C$. Then $c-b\in G\subseteq B$. Therefore, $c\in B\cap C=0$ and $h(c)=h(0)$. This is a contradiction.

3. For any element $m\in M$, the module $mG_{A}$ is a Goldie-radical module, since $mG_{A}$ is a homomorphic image of the Goldie-radical module $G$. Therefore, $mG\subseteq G(M)$ and $MG\subseteq G(M)$.

4. By 3, $XG=0$. Let us assume that $x\in X$ and $xh(B)=0$ for some essential right ideal $h(B)$, where $B=h^{-1}(h(B))$ is the complete pre-image of $h(B)$ in the ring $A$. By 2), $B$ is an essential right ideal of the ring $A$. Then $xB=0$ and $x\in\operatorname{Sing}X=0$. Therefore, $X$ is a non-singular $h(A)$-module. The remaining part of 4 is directly verified.

5. Let $R$ be one of the rings $A$, $h(A)$ and $M$ a right $R$-module. By Lemma 1(4), the module $M$ is injective if and only if $M$ is injective relative to the module $R_{R}$. Now the assertion follows from 4.

6. The assertion follows from 4.

7. Let $\mathcal{M}$ be the set of all submodules of the module $X$ which are direct sums of modules each of them is isomorphic to a non-zero right ideal of the ring $A$. The set $\mathcal{M}$ is not empty by Lemma 3.1. There exists a partial order in $\mathcal{M}$ such that for any $M,M^{\prime}\in\mathcal{M}$, the relation $M\lneqq M^{\prime}$ is equivalent to the property that $M^{\prime}=M\oplus N$ for some $N\in\mathcal{M}$. By the Zorn lemma, the set $\mathcal{M}$ contains at least one maximal element $K$.

Let us assume that $K$ is not an essential submodule of the module $X$. Then there exists a non-zero submodule $L$ of the non-singular module $X$ with $K\cap L=0$. By Lemma 3.1, there exists a non-zero right ideal $B$ of the ring $A$ such that the module $B_{A}$ is isomorphic to some submodule $L^{\prime}$ of the module $L$. This contradicts to the property that $K$ is a maximal element of the set $\mathcal{M}$.

8. Since the ring $A$ is right finite-dimensional, every non-zero right ideal of the ring $A$ is an essential extension of a finite direct sum of non-zero uniform right ideals. Now the assertion follows from 7.

9. The assertion follows from 6 and 8.  $\square$

Remark 3.3. Let $M$ be an automorphism-invariant non-singular module. In [3, Theorem 3, Theorem 6(ii)] Er, Singh and Srivastava proved that $M=X\oplus Y$, where $X$ is a quasi-injective non-singular module, $Y$ is an automorphism-invariant non-singular square-free module, the modules $X$ and $Y$ are injective relative to each other, $\operatorname{Hom}(X,Y)=0=\operatorname{Hom}(Y,X)$ and $\operatorname{Hom}(D_{1},D_{2})=0$ for any two submodules $D_{1}$, $D_{2}$ of the module $Y$ with $D_{1}\cap D_{2}=0$. In addition, for any set $\{K_{i}\,|\,i\in I\}$ of closed submodules of $Y$, the submodule $\sum_{i\in I}K_{i}$ is an automorphism-invariant module.

Remark 3.4. Let $M$ be a direct sum of CS modules $M_{i}$, $i\in I$. In [11, Corollary 15] Lee and Zhou proved that $M$ is a quasi-injective module if and only if $M$ is an automorphism-invariant module.

Remark 3.5. Let $A$ be a ring with right Goldie radical $G(A_{A})$.

In [9, Theorem 3.4] Kutami and Oshiro proved that the factor ring $A/G(A_{A})$ is a right strongly semiprime ring if and only if every non-singular quasi-injective right $A$-module is injective.

In [9, Theorem 3.8] Kutami and Oshiro proved that the factor ring $A/G(A_{A})$ is a semiprime right Goldie ring if and only if every direct sum of non-singular quasi-injective right $A$-modules is quasi-injective.

Lemma 3.6. Let $A$ be a ring with right Goldie radical $G(A_{A})$ and $M$ an automorphism-invariant non-singular right $A$-module which is an essential extension of a direct sum of uniform modules.

1. 1)

$M$* is an essential extension of some quasi-injective non-singular module $K$ which is direct sum of uniform modules closed in $M$.* 2. 2)

If the factor ring $A/G(A_{A})$ is a right strongly semiprime ring, then $M$ is an injective module.

Proof. 1. By Remark 3.3, $M=X\oplus Y$, where $X$ is a quasi-injective module, $Y$ is an automorphism-invariant square-free module. Therefore, we can assume that $M$ is an automorphism-invariant square-free module. Since $M$ is an essential extension direct sum of uniform submodules, $M$ is an essential extension of some module $K$ which is the direct sum of uniform closed submodules $K_{i}$ of $M$, $i\in I$. By Remark 3.3, $K$ is an automorphism-invariant module. Since every uniform module is a CS module, $K$ is a quasi-injective module by Remark 3.4.

2. By 1, $M$ is an essential extension of some quasi-injective non-singular module $K$. By Remark 3.5, $K$ is an injective essential submodule of the module $M$. Therefore, $K$ is an essential direct summand of the module $M$. Then $M=K$ and $M$ is an injective module.  $\square$

Lemma 3.7. Let $A$ be a ring with right Goldie radical $G(A_{A})$ and $M$ an automorphism-invariant non-singular right $A$-module. If the factor ring $A/G(A_{A})$ is a semiprime right Goldie ring, then $M$ is an injective module.

Proof. By Lemma 3.2(9), $M$ is an essential extension of a direct sum of uniform modules. In addition, the semiprime right Goldie ring $A/G(A_{A})$ is a right strongly semiprime ring [6]. By Lemma 3.6(2), $M$ is an injective module.  $\square$

Remark 3.8. The completion of the proof of Theorem 1.5. In Theorem 1.5, the implications 3) $\Rightarrow$ 2) $\Rightarrow$ 1) are obvious.

1) $\Rightarrow$ 3). Let $M$ be the direct sum of automorphism-invariant non-singular right $A$-modules $M_{i}$, $i\in I$. By Lemma 3.7, each of the modules $M_{i}$ is injective. By Remark 3.5, $M$ is a quasi-injective module. By Remark 3.5, $M$ is an injective module.  $\square$