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A submodule N of a module M is δ-small in M if N+X≠M for any proper submodule X of M with M∕X singular. A projective δ-cover of a module M is a projective module P with an epimorphism to M whose kernel is δ-small in P. A module M is called δ-semiperfect if every factor module of M has a projective δ-cover. In this paper, we prove various properties, including a structure theorem and several characterizations, for δ-semiperfect modules. Our proofs can be adapted to generalize several results of Mares [8] and Nicholson [11] from projective semiperfect modules to arbitrary semiperfect modules. [ABSTRACT FROM AUTHOR]

It is proved that a semiperfect module is lifting if and only if it has a projective cover preserving direct summands. Three corollaries are obtained: (1) every cyclic module over a ring R is lifting if and only if every cyclic R-module has a projective cover preserving direct summands; (2) a ring R is artinian serial with Jacobson radical square-zero if and only if every (2-generated) R-module has a projective cover preserving direct summands; (3) a ring R is a right (semi-)perfect ring if and only if (cyclic) lifting R-module has a projective cover preserving direct summands, if and only if every (cyclic) R-module having a projective cover preserving direct summands is lifting. It is also proved that every cyclic module over a ring R is ⊕-supplemented if and only if every cyclic R-module is a direct sum of local modules. Consequently, a ring R is artinian serial if and only if every left and right R-module is a direct sum of local modules. [ABSTRACT FROM AUTHOR]

Let R be a ring with identity. The unitary Cayley graph of R is the simple graph with vertex set R , where two distinct vertices x and y are linked by an edge if and only if x − y is a unit of R. A graph is said to be planar if it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In this paper, we completely characterize the rings whose unitary Cayley graphs are planar. [ABSTRACT FROM AUTHOR]

A module is called a -module if, whenever and are submodules of with and is a homomorphism with , we have . The class of -modules contains the -modules as well as the dual-square-free (DSF) modules. Furthermore, a -module is called pseudo-discrete if is also a lifting module. In this paper, we study the -, the DSF, and the pseudo-discrete modules, and show that a pseudo-discrete module is clean iff it has the finite exchange property iff it has the full exchange property. [ABSTRACT FROM AUTHOR]

AnR-moduleM is called aC4-module if, wheneverA1andA2are submodules ofMwithandis anR-homomorphism with, we have. This notion is a non-trivial generalization of both theC3-modules and the square-free modules. Furthermore, aC4-module is called pseudo-continuous if it is also aCS-module. In this paper, we give a thorough study of theC4- and the pseudo-continuous modules. [ABSTRACT FROM AUTHOR]

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 . Here we carry out a study of the rings whose cyclic modules are -modules. The motivation is the observation that a ring is semisimple artinian if and only if every -generated right -module is a -module. Many basic properties are obtained for the rings whose cyclics are -modules, and some structure theorems are proved. For instance, it is proved that a semiperfect ring has all cyclics -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 -modules if and only if it is a direct product of a semisimple artinian ring, a strongly regular ring and a matrix ring over a strongly regular ring. Applications to the rings whose -generated modules are -modules, and the rings whose cyclics are ADS or quasi-continuous are addressed. [ABSTRACT FROM AUTHOR]

A right moduleMover a ringRis called feebly Baer if, wheneverxa = 0 withx ∈ Manda ∈ R, there exists e2 = e ∈ Rsuch that xe = 0 and ea = a. The ringRis called feebly Baer if RRis a feebly Baer module. These notions are motivated by the commutative analog discussed in a recent paper by Knox, Levy, McGovern, and Shapiro [6]. Basic properties of feebly Baer rings and modules are proved, and their connections with von Neumann regular rings are addressed. [ABSTRACT FROM AUTHOR]

A module is defined to be an automorphism-invariant module if it is invariant under automorphisms of its injective hull. Quasi-injective modules and, more generally, pseudo-injective modules are all automorphism-invariant. Here we develop basic properties of these modules, and discuss when an automorphism-invariant module is quasi-injective or injective. Some known results on quasi-injective and pseudo-injective modules are extended. [ABSTRACT FROM AUTHOR]

This article is a continuation of the work in [3] on clean index of rings. Motivated by recent work [5] on uniquely clean rings, the clean index of a ring was introduced in [3] to study the structure of rings, where arbitrary rings of clean indices 1, 2, and 3, respectively, are characterized. In particular, clean rings of clean indices 1, 2, and 3 are determined. In this article, we characterize the rings of clean index 4. As applications, clean rings of clean indices 4, 5, 6 and 7 are completely determined. [ABSTRACT FROM PUBLISHER]

A ring is called clean (resp., uniquely clean) if each of its elements can be (resp., uniquely) expressed as the sum of an idempotent and a unit. Motivated by recent work on uniquely clean rings in [6], we introduce the clean index of a ring R. For a ∈ R, let ℰ(a) = {e ∈ R: e 2 = e, a − e ∈ U(R)} where U(R) is the group of units of R and the clean index of R, denoted in(R), is defined by in(R) = sup{|ℰ(a)|: a ∈ R}. Thus, R is uniquely clean if and only if R is clean with in(R) = 1. So far, uniquely clean rings are the only clean rings whose structure is fully understood (see [6]). In this article, we characterize the (arbitrary) rings of clean indices 1, 2, 3 and determine the abelian rings of finite clean index. Applications to semipotent rings, semiprime rings, and clean rings are discussed. [ABSTRACT FROM PUBLISHER]

Let M and N be two modules over a ring R. The concern is about the four substructures of homR(M, N): the Jacobson radical J[M, N], the singular ideal Δ[M, N], the co-singular ideal ∇[M, N] and the total Tot[M, N]. One natural question is to characterize when the total is equal to one or more of the other structures. We review some known results and prove several new results towards this question and, as consequences, give answers to some related questions. [ABSTRACT FROM AUTHOR]

Abstract: A new characterization of von Neumann regular rings is obtained, in terms of simple 0-multiplication of matrices, and is used to establish the natural connections between von Neumann regular rings and feebly Baer modules and rings. [Copyright &y& Elsevier]

A ring R is called left morphic if, for each a ∈ R, R/Ra ≅ l(a) or equivalently there exists b ∈ R such that Ra = l(b) and l(a) = Rb, where l(a) and l(b) denote the left annihilators of a and b in R, respectively. Motivated by recent work on left morphic rings, we study the rings R satisfying the property that for each 0 ≠ a ∈ R there exists a unique b ∈ R such that Ra = l(b) and l(a) = Rb. These rings are completely determined in this paper. We also completely determine the rings R with the property that for each 0 ≠ a ∈ R there exists a unique b ∈ R such that Ra = l(b). [ABSTRACT FROM AUTHOR]

Abstract: Let M and N be two modules over a ring R. Recent works by Kasch, Schneider, Beidar, Mader, and others have shown that some of the ring and module theory can be carried out in the context of . The study of substructures of such as the radical, the singular and co-singular ideals and the total has raised new questions for research in this area. This paper is a continuation of study of these substructures, focusing on when the total is equal to the radical, as well as their connections with (semi)regularity of . New results obtained include necessary and sufficient conditions for the total to equal the radical, a description of the maximal regular sub-bimodule of , the existence of the maximal semiregular ideal of a ring, and answers to a number of existing open questions. [Copyright &y& Elsevier]

Let S and R be rings. The objective is to study the bimodule [sub s]N[sub R] satisfying the annihilator conditions l[sub N](r[sub R](x)) = Sx for all x ∈ N. This approach will clearly show how the ring R or the module N[sub R] is connected to the properties of the ring S through the annhilator condition. Specializing to the particular bimodule [sub R]R[sub R] or [sub End](N)N[sub R], we obtain some new results and known results as corollaries. [ABSTRACT FROM AUTHOR]

Let R be a ring. For two fixed positive integers m and n, a right R-module M is called (m, n)-injective if every right R-homomorphism from an n-generated submodule of R[SUPm] to M extends to one from R[SUPm] to M. This definition unifies several definitions on generalizations of injectivity of modules. The aim of this paper is to investigate properties of the (m,n)-injective modules. Various results are developed, many extending known results. [ABSTRACT FROM AUTHOR]

For a semisimple module M over a ring R with R / J ( R ) Boolean, every endomorphism of M is a sum of an idempotent endomorphism and a locally nilpotent endomorphism. As a consequence, it is proved that, for a vector space V over a division ring D , every linear transformation of V is a sum of an idempotent linear transformation and a locally nilpotent linear transformation if and only if D ≅ F 2 . This can be seen as an answer to the “local” version of a question raised by Breaz et al. in [1] on nil-cleanness of the ring of linear transformations of an infinite dimensional vector space. [ABSTRACT FROM AUTHOR]