相关论文: Indecomposable modules and Gelfand rings
An element $a$ in a ring $R$ is strongly J-clean if it is the sum of an idempotent and an element in the Jacobson radical that commutes. We characterize the strongly J-clean $2\times 2$ matrices over 2-projective-free non-commutative rings.
In this article, we introduce a new graph theoretic structure associated with a finite commutative ring, called nil clean divisor graph. For a ring $R$, nil clean divisor graph is denoted by $G_N(R)$, where the vertex set is $\{x\in R\,:\,…
A ring $R$ is called strongly clean if every element of $R$ is the sum of a unit and an idempotent that commute. By {\rm SRC} factorization, Borooah, Diesl, and Dorsey \cite{BDD051} completely determined when ${\mathbb M}_n(R)$ over a…
We prove that if an involution in a ring is the sum of an idempotent and a nilpotent then the idempotent in this decomposition must be 1. As a consequence, we completely characterize weakly nil-clean rings introduced recently in [Breaz,…
In this paper we review and study $R$-modules $M$ for which $S = End_R(M)$ is commutative. For this, we define the concept of center of modules which is a natural generalization of the center of rings. The properties of center of modules,…
Non-commutative Henselian rings are defined and it is shown that a local ring which is complete and separated in the topology defined by its maximal ideal is Henselian provided that it is almost commutative.
It is proved that every commutative ring whose RD-injective modules are $\Sigma$-RD-injective is the product of a pure semi-simple ring and a finite ring. A complete characterization of commutative rings for which each artinian…
The structure of cyclically pure injective modules over a commutative ring $R$ is investigated and several characterizations for them are presented. In particular, we prove that a module $D$ is cyclically pure injective if and only if $D$…
A module is called absolutely indecomposable if it is directly indecomposable in every generic extension of the universe. We want to show the existence of large abelian groups that are absolutely indecomposable. This will follow from a more…
The main result is Theorem: Let A be an R-algebra, mu, lambda be cardinals such that |A|<=mu=mu^{aleph_0}<lambda<=2^mu. If A is aleph_0-cotorsion-free or A is countably free, respectively, then there exists an aleph_0-cotorsion-free or a…
A ring is called clean if every element is the sum of an invertible element and an idempotent. This paper investigates the cleanness of AW*-algebras. We prove that all finite AW*-algebras are clean, affirmatively solving a question posed by…
Given a significative class $F$ of commutative rings, we study the precise conditions under which a commutative ring $R$ has an $F$-envelope. A full answer is obtained when $F$ is the class of fields, semisimple commutative rings or…
We define topologically semiperfect (complete, separated, right linear) topological rings and characterize them by equivalent conditions. We show that the endomorphism ring of a module, endowed with the finite topology, is topologically…
This study explores in-depth the structure and properties of the so-called {\it strongly $\Delta$-clean rings}, that is a novel class of rings in which each ring element decomposes into a sum of a commuting idempotent and an element from…
In this paper, we construct indecomposable integrally closed modules of arbitrary rank over a two-dimensional regular local ring. The modules are quite explicitly constructed from a given complete monomial ideal. We also give structural and…
It is proven that every commutative semihereditary Bezout ring in which any regular element is Gelfand (adequate), is an elementary divisor ring.
Using the concept of ring of Gelfand range 1 we proved that a commutative Bezout domain is an elementary divisor ring iff it is a ring of Gelfand range 1. Obtained results give a solution of problem of elementary divisor rings for different…
We systematically study those rings whose non-units are a sum of an idempotent and a nilpotent. Some crucial characteristic properties are completely described as well as some structural results for this class of rings are obtained. This…
In this article, we construct integrally closed modules of rank two over a two-dimensional regular local ring. The modules are explicitly constructed from a given complete monomial ideal with respect to a regular system of parameters. Then…
We prove that the only separable commutative ring-objects in the stable module category of a finite cyclic p-group G are the ones corresponding to subgroups of G. We also describe the tensor-closure of the Kelly radical of the module…