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A right R -module M is called a U -module if, whenever A and B are submodules of M with A ≅ B and A ∩ B = 0 , there exist two direct summands K and T of M such that A ⊆ ess K , B ⊆ ess T and K ⊕ T ⊆ ⊕ M. The class of U -modules is a strict and simultaneous generalization of quasi-continuous, square-free and automorphism-invariant modules. In this paper we study the rings whose cyclics are U -modules, extending many of the known results on the subject and providing new ones. [ABSTRACT FROM AUTHOR]

The aim of the paper is to characterize the equation 1 + Δ (R) = U (R) , where J (R) ⊆ Δ (R) = { x ∈ R : x + u ∈ U (R)   for all   u ∈ U (R) } that is the largest Jacobson radical subring of R and U (R) is the set of invertible elements of a ring R. We show that this equation is closely related to U J -rings and rings whose elements can be written as the sum of an idempotent and an element from Δ (R). After presenting several characterizations and properties of this equation, we consider the rings satisfying the equation 1 + Δ (R) = U (R) within many well-studied classes of rings. Finally, we close the paper with group rings. [ABSTRACT FROM AUTHOR]

The object of this article is associate to automorphism-invariant modules that are invariant under any automorphism of their injective hulls with cyclic modules and cyclic modules have cyclic automorphism-invariant hulls. The study of the first sequence allows us to characterize rings whose cyclic right modules are automorphism-invariant and to show that if R is a right Köthe ring, then R is an Artinian principal left ideal ring in case every cyclic right R-module is automorphism-invariant. The study of the second sequence leads us to consider a generalization of hypercyclic rings that are each cyclic R-module has a cyclic automorphism-invariant hull. Such rings are called right a-hypercyclic rings. It is shown that every right a-hypercyclic ring with Krull dimension is right Artinian. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to introduce and study the dual concepts of weakly automorphism invariant modules and essential tightness. These notions are non-trivial generalizations of both weakly projectivity, dual automorphism invariant property and cotightness. We obtain certain relations between weakly projective modules, weakly dual automorphism invariant modules and superfluous cotight modules. It is proved that: (1) for right perfect rings, every module is a direct summand of a weakly dual automorphism invariant module and (2) weakly dual automorphism invariant modules are precisely superfluous cotight modules. [ABSTRACT FROM AUTHOR]

A ring R is called eversible if every left zero-divisor in R is also a right zero-divisor and conversely. This class of rings is a natural generalization of reversible rings. It is shown that every eversible ring is directly finite, and a von Neumann regular ring is directly finite if and only if it is eversible. We give several examples of some important classes of rings (such as local, abelian) that are not eversible. We prove that R is eversible if and only if its upper triangular matrix ring T n (R) is eversible, and if M n (R) is eversible then R is eversible. [ABSTRACT FROM AUTHOR]

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]

Let R be a commutative Noetherian ring with non-zero identity, 픞 an ideal of R and M a finitely generated R-module. We assume that N is a weakly Laskerian R-module and r is a non-negative integer such that the generalized local cohomology module is weakly Laskerian for all i < r. Then we prove that is also weakly Laskerian and so is finite. Moreover, we show that if s is a non-negative integer such that is weakly Laskerian for all i, j ≥ 0 with i ≤ s, then is weakly Laskerian for all i ≤ s and j ≥ 0. Also, over a Gorenstein local ring R of finite Krull dimension, we study the question when the socle of is weakly Laskerian? [ABSTRACT FROM AUTHOR]

Let M be a right R-module and N ∈ σ[M]. A submodule K of N is called δ-M-small if, whenever N = K + X with N/X M-singular, we have N = X. N is called a δ-M-small module if N≅ K, K is δ-M-small in L for some K, L ∈ σ[M]. In this article, we prove that if M is a finitely generated self-projective generator in σ[M], then M is a Noetherian QF-module if and only if every module in σ[M] is a direct sum of a projective module in σ[M] and a δ-M-small module. As a generalization of a Harada module, a module M is called a δ-Harada module if every injective module in σ[M] is δM-lifting. Some properties of δ-Harada modules are investigated and a characterization of a Harada module is also obtained. [ABSTRACT FROM AUTHOR]