Self-injective Rings Research Articles

Commutative rings whose proper ideals decompose into local modules

For a commutative local ring R, we know that every ideal is a direct sum of local modules if and only if the unique maximal ideal of R is of the form , where and are principal ideal rings and for all γ’s. This motivates us to determine a commutative ring, not necessarily local, in which every (proper) ideal is a direct sum of local modules. We prove that such a ring is exactly a finite direct product of local rings R with the unique maximal ideal in the above mentioned form. Moreover, we characterize more precisely certain commutative rings such as Noetherian rings, reduced rings, Rickart rings, semi-Artinian rings and self-injective rings, in which every (proper) ideal is a direct sum of local modules.

Higher Dualizability and Singly-Generated Grothendieck Categories

Let k be a field. We show that locally presentable, k-linear categories $${\mathcal {C}}$$ dualizable in the sense that the identity functor can be recovered as $$\coprod _i x_i\otimes f_i$$ for objects $$x_i\in {\mathcal {C}}$$ and left adjoints $$f_i$$ from $${\mathcal {C}}$$ to $$\mathrm {Vect}_k$$ are products of copies of $$\mathrm {Vect}_k$$ . This partially confirms a conjecture by Brandenburg, the author and T. Johnson-Freyd. Motivated by this, we also characterize the Grothendieck categories containing an object x with the property that every object is a copower of x: they are precisely the categories of non-singular injective right modules over simple, regular, right self-injective rings of type I or III.

The radius of unit graphs of rings

<abstract><p>Let $ R $ be a ring with nonzero identity. The unit graph of $ R $ is a simple graph whose vertex set is $ R $ itself and two distinct vertices are adjacent if and only if their sum is a unit of $ R $. In this paper, we study the radius of unit graphs of rings. We prove that there exists a ring $ R $ such that the radius of unit graph can be any given positive integer. We also prove that the radius of unit graphs of self-injective rings are 1, 2, 3, $ \infty $. We classify all self-injective rings via the radius of its unit graph. The radius of unit graphs of some ring extensions are also considered.</p></abstract>

On the Restricted Minimum Condition for Rings

Generalizing Artinian rings, a ring R is said to have right restricted minimum condition ( $${\mathrm{r.RMC}}$$ , for short) if R/A is an Artinian right R-module for any essential right ideal A of R. It is asked in Jain et al. [Cyclic Modules and the Structure of Rings, Oxford University Press, Oxford, 2012, 3.17 Questions (2)] that (i) Is a left self-injective ring with $${\mathrm{r.RMC}}$$ quasi-Frobenius? (ii) Whether a serial ring with $${\mathrm{r.RMC}}$$ must be Noetherian? We carry out a study of rings with $${\mathrm{r.RMC}}$$ and determine when a right extending ring has $${\mathrm{r.RMC}}$$ in terms of rings $${\begin{bmatrix} S&{}\quad M\\ 0&{}\quad R\end{bmatrix}}$$ such that S is right Artinian, $$M_{Q}$$ is semisimple ( $$Q={\mathrm{Q}}(R)$$ ) and R is a semiprime ring with Krull dimension 1. We proved that a left self-injective ring R with $${\mathrm{r.RMC}}$$ is quasi-Frobenius if and only if $$\hbox {Z}_{r}(R) = \hbox {Z}_{l}(R)$$ if and only if $$\hbox {Z}_{r}(R)$$ is a finitely generated left ideal and $${\mathrm{N}}(R)\cap {\mathrm{Soc}}(R_{R})$$ is a finitely generated right ideal. Right serial rings with $${\mathrm{r.RMC}}$$ are studied and proved that a non-singular serial ring has $${\mathrm{r.RMC}}$$ if and only if it is a left Noetherian ring. Examples are presented to describe our results and to show that $$\mathrm{RMC}$$ is not symmetric for a ring.

Triangular matrix rings of selfinjective rings

A module M is said to be generalized extending if for every submodule there exists a direct summand D of M containing N such that D/N is a singular module. In this note we prove that a ring R is right self-injective if and only if the triangular ring is right generalized extending. This answers a question which was raised in A. Akalan, G.F. Birkenmeier, A. Tercan, Characterizations of extending modules and -extending generalized triangular matrix rings, Commun. Algebra 40 (2012), 1069–1085.

When proper cyclics are homomorphic image of injectives

Quasi-Frobenius rings are precisely rings over which any right module is a homomorphic image of an injective module. We investigate the structure of rings whose proper cyclic right modules are homomorphic image of injectives. The class of such rings properly contains that of right self-injective rings. We obtain some structure theorems for rings satisfying the said property and apply them to the Artin algebra case: It follows that an Artin algebra with this property is Quasi-Frobenius.

Representations of Quivers over Artinian Rings

Given a unitary artinian ring R and a finite acyclic quiver Q, let Λ := RQ be the path ring of Q over R. Then Gorenstein-projective Λ-modules are exactly the separated monic representations of Q over R which satisfy the local Gorenstein-projective condition. We denote by smon(Q,R) the category of all finitely generated separated monic representations of Q over R, and $\mathcal {G}p({\varLambda })$ the category of all finitely generated Gorenstein-projective Λ-modules. If R is a selfinjective ring, then $\mathcal {G}p({\varLambda })=\text {smon}{\it (Q, R) }$ . As an application, if R is a commutative uniserial selfinjective ring of length 2 (here it means that as a regular module R is uniserial with length 2), let 0 ≠ a ∈radR and $\bar {R}=R/\text {rad}\textit {R}$ , our main result says that there is a full functor $H: \mathcal {G}p({\varLambda }) \to \text {mod} {\bar {R}Q}$ which induces a bijection between the indecomposable non-projetive Gorenstein-projective Λ-modules and the indecomposable $ \bar {R}Q$ -modules. Moreover, in this case, each Gorenstein-projective Λ-module is strongly Gorenstein-projective.

On a generalization of self-injective rings

In this work the notion of left (right) self-injective ring is generalized. We consider rings that are direct sum of injective module and semisimple module as a left (respectively, right) module above itself. We call such rings left (right) semi-injective and research their properties with the help of two-sided Peirce decomposition of the ring. The paper contains the description of left noetherian left semi-injective rings. It is proved that any such ring is a direct product of (two-sided) self-injective ring and several quotient rings (of special kind) of rings of upper triangular matrices over skew fields. From this description it follows that for left semi-injective rings we have the analogue of the classical result for self-injective rings. Namely, if a ring is left noetherian and left semi-injective then this ring is also right semi-injective and two-sided artinian.

On a Generalization of Self-Injective Rings

On a Generalization of Self-Injective Rings

On self-injectivity of the f-ring Frm(𝓟(ℝ), L)

Abstract Let 𝓕𝓟L := Frm(𝓟(ℝ), L). We show that if L is a P-frame then 𝓕𝓟L is an ℵ0-self-injective ring. We prove that a zero-dimensional frame L is extremally disconnected if and only if 𝓕𝓟L is a self-injective ring. Finally, it is shown that 𝓕𝓟L is a Baer ring if and only if 𝓕𝓟L is a continuous ring if and only if 𝓕𝓟L is a complete ring if and only if 𝓕𝓟L is a CS-ring.

On subrings of the form $I+\mathbb {R}$ of $C(X)$

In this article we observe that $C(X)$ is integral over $I+\mathbb R$ if and only if $I$ is of the form of a finite intersection of real maximal ideals. The integral closure of a subring $I+\mathbb R$ is investigated and it turns out that $I+\mathbb R$ is integrally closed if and only if $I$ is semiprime and $\theta (I)$ is connected. We show that $I^u+\mathbb R$, where $I^u$ is the uniform closure of the ideal $I$, is always a ring of quotients of $I+\mathbb R$ and $C(X)$ is a ring of quotients of $I+\mathbb R$ if and only if $I$ is either essential (large) or a maximal ideal generated by an idempotent. Maximal subrings of the form $I+\mathbb R$ of $C(X)$ are characterized and it is shown that $[M^p]^u+\mathbb R$ (which is isomorphic to $C(X\cup \{p\})$, where $X\cup \{p\}$ is a subspace of $\beta X$) is not contained in a proper maximal subring of $C(X)$. We have observed that for each subring of the form $I+\mathbb R$ of $C(X)$, the sum of every two prime ideals of $I+\mathbb R$ is prime or all of $I+\mathbb R$ if and only if $X$ is an $F$-space. Ideals $I$ are characterized for which $I+\mathbb R$ is a regular ring and for every $z$-ideal $I$, we have shown that the sum of every two $z$-ideals of $I+\mathbb R$ is a $z$-ideal or all of $I+\mathbb R$. Finally, some algebraic and topological properties are given for which $I+\mathbb R$ is a self-injective ring or a uem-ring.

ℵ0-Injective group rings

Recall that a ring [Formula: see text] is called right [Formula: see text]-injective if every homomorphism from a countably generated right ideal of [Formula: see text] to [Formula: see text] can be extended to a homomorphism from [Formula: see text] to [Formula: see text]. These rings are not only a natural generalization of self-injective rings but also strongly connected with regularities of rings. Let [Formula: see text] be the group ring of a group [Formula: see text] over a ring [Formula: see text]. It is proved that [Formula: see text] is right [Formula: see text]-injective if and only if (i) [Formula: see text] is right [Formula: see text]-injective; (ii) [Formula: see text] is finite; (iii) for each countably generated right ideal [Formula: see text] of [Formula: see text], any right [Formula: see text]-homomorphism from [Formula: see text] to [Formula: see text] can be extended to a right [Formula: see text]-homomorphism from [Formula: see text] to [Formula: see text].

On injectivity of the ring of real-valued continuous functions on a frame

We give characterizations of $P$-frames and extremally disconnected $P$-frames based on ring-theoretic features of the ring of continuous real- valued functions on a frame $L$, i.e. $\mathcal RL$. It is shown that $L$ is a $P$-frame if and only if $\mathcal RL$ is an $\aleph_0$-self-injective ring. Consequently for pseudocompact frames if $\mathcal RL$ is $\aleph_0$-self-injective, then $L$ is finite. We also prove that $L$ is an extremally disconnected $P$-frame iff $\mathcal{R}L$ is a self-injective ring iff $\mathcal{R}L$ is a Baer regular ring iff $\mathcal{R}L$ is a continuous regular ring iff $\mathcal{R}L$ is a complete regular ring.

On endo-semiprime and endo-cosemiprime modules

In this paper, we study the notions of endo-semiprime and endo-cosemiprime modules and obtain some related results. For instance, we show that in a right self-injective ring $R$, all nonzero ideals of $R$ are endo-semiprime as right (left) $R$-modules if and only if $R$ is semiprime. Also, we prove that both being endo-semiprime and being are Morita invariant properties.

Divisibility on chains of submodules

ABSTRACTAn R-module M is said to satisfy ACCd (resp. DCCd) on submodules if for every ascending (resp. descending) chain {Mi} of submodules of M, (resp. ) for some for i≫0. A nonzero module with ACCd or DCCd on submodules contains an essential submodule which is a direct sum of uniform submodules almost all noetherian. We show that if M is a finitely generated self-projective and self-injective R-module with ACCd or DCCd on submodules, then M is a finite direct sum of uniform submodules. It is shown that a regular right self-injective ring with ACCd or DCCd on right ideals must be semisimple artinian. We also prove that if M is a nonzero nonsingular module over a right noetherian ring and E(M)(ℕ) satisfies ACCd or DCCd on submodules, then M is semisimple. Next we consider some conditions for modules with ACCd (resp. DCCd) on submodules to satisfy ACC (resp. DCC) on some families of prime submodules. Finally, we show that a commutative ring with DCCd on ideals has dimension at most 1.

Rings whose cyclics are C3-modules

It is well known that if every cyclic right module over a ring is injective, then the ring is semisimple artinian. This classical theorem of Osofsky promoted a considerable interest in the rings whose cyclics satisfy a certain generalized injectivity condition, such as being quasi-injective, continuous, quasi-continuous, or [Formula: see text]. Here we carry out a study of the rings whose cyclic modules are [Formula: see text]-modules. The motivation is the observation that a ring [Formula: see text] is semisimple artinian if and only if every [Formula: see text] -generated right [Formula: see text]-module is a [Formula: see text]-module. Many basic properties are obtained for the rings whose cyclics are [Formula: see text]-modules, and some structure theorems are proved. For instance, it is proved that a semiperfect ring has all cyclics [Formula: see text]-modules if and only if it is a direct product of a semisimple artinian ring and finitely many local rings, and that a right self-injective regular ring has all cyclics [Formula: see text]-modules if and only if it is a direct product of a semisimple artinian ring, a strongly regular ring and a [Formula: see text] matrix ring over a strongly regular ring. Applications to the rings whose [Formula: see text]-generated modules are [Formula: see text] -modules, and the rings whose cyclics are ADS or quasi-continuous are addressed.

Recent development of Faith conjecture

Is a semiprimary right self-injective ring a quasi-Frobenius ring? Almost half century has passed since Faith raised this problem. He first conjectured “No” in his book Algebra II. Ring Theory in 1976, but changing his mind, he conjectured “Yes” in his article “When self-injective rings are QF: a report on a problem” in 1990. In this paper, we describe recent studies of this problem based on authors works and raise related problems.

Almost relative injective modules

The concept of a module $M$ being almost $N$-injective, where $N$ is some module, was introduced by Baba (1989). For a given module $M$, the class of modules $N$, for which $M$ is almost $N$-injective, is not closed under direct sums. Baba gave a necessary and sufficient condition under which a uniform, finite length module $U$ is almost $V$-injective, where $V$ is a finite direct sum of uniform, finite length modules, in terms of extending properties of simple submodules of $V$. Let $M$ be a uniform module and $V$ be a finite direct sum of indecomposable modules. Some conditions under which $M$ is almost $V$-injective are determined, thereby Baba's result is generalized. A module $M$ that is almost $M$-injective is called an almost self-injective module. Commutative indecomposable rings and von Neumann regular rings that are almost self-injective are studied. It is proved that any minimal right ideal of a von Neumann regular, almost right self-injective ring, is injective. This result is used to give an example of a von Neumann regular ring that is not almost right self-injective.

C3-Modules

One of the continuity conditions identified by Utumi on self-injective rings is the C3-condition, where a module M is called a C3-module if whenever A and B are direct summands of M and A ∩ B=0, then A ⊕ B is a summand of M. In addition to injective and direct-injective modules, the class of C3-modules includes the semisimple, continuous, indecomposable and regular modules. Indeed, every commutative ring is a C3-ring. In this paper we provide a general and unified treatment of the above mentioned classes of modules in terms of the C3-condition, and establish new characterizations of several well known classes of rings.

When is a Sum of Annihilator Ideals an Annihilator Ideal?

We call a ring R a right SA-ring if for any ideals I and J of R there is an ideal K of R such that r(I) + r(J) = r(K). This class of rings is exactly the class of rings for which the lattice of right annihilator ideals is a sublattice of the lattice of ideals. The class of right SA-rings includes all quasi-Baer (hence all Baer) rings and all right IN-rings (hence all right selfinjective rings). This class is closed under direct products, full and upper triangular matrix rings, certain polynomial rings, and two-sided rings of quotients. The right SA-ring property is a Morita invariant. For a semiprime ring R, it is shown that R is a right SA-ring if and only if R is a quasi-Baer ring if and only if r(I) + r(J) = r(I ∩ J) for all ideals I and J of R if and only if Spec(R) is extremally disconnected. Examples are provided to illustrate and delimit our results.