Related papers: Indecomposable modules and Gelfand rings

We prove that indecomposable $\Sigma$-pure-injective modules for a string algebra are string or band modules. The key step in our proof is a splitting result for infinite-dimensional linear relations.

Recall that an element $x\in R$ is {\bf complemented} if there is a $y\in R$ such that $xy = 0$ and $x + y \in {\rm reg}(R)$. In a recent article [1], the authors investigated those rings for which every non-nilpotent element is…

We introduce the concept of a weak nil clean ring, a generalization of nil clean ring, which is nothing but a ring with unity in which every element can be expressed as sum or difference of a nilpotent and an idempotent. Further if the…

According to an old result of Albert and Muckenhoupt, the commutators in the endomorphism ring of a finite dimensional vector space are precisely the elements of trace zero. We replace the finite dimensional vector space with a complex of…

We study the construction and properties of modules whose endomorphism rings have a unique two-sided maximal ideal.

We prove conditions ensuring that a Lie ideal or an invariant additive subgroup in a ring contains all additive commutators. A crucial assumption is that the subgroup is fully noncentral, that is, its image in every quotient is noncentral.…

In this paper, strongly clean ring defined by W. K. Nicholson in 1999 has been generalized to n-strongly clean, {\Sigma}-strongly clean and with the help of example it has been shown that there exists a ring, which is n-strongly clean and…

Let (R,m) be a noetherian local ring and let $\mathcal{C}$ be the class of all R-modules M which possess a reflexive submodule U such that M/U is finitely generated. For every R-module $M\in \mathcal{C}$ the canonical embedding $\varphi:…

We give a characterization of indecomposable exceptional modules over finite dimensional gentle algebras. As an application, we study gentle algebras arising from an unpunctured surface and show that a class of indecomposable modules…

In Finite Group Modular Representation Theory, the basic objects are the indecomposable and simple modules. This paper offers a new classification of these objects that refines the Green Theory Classification of indecomposable and simple…

The aim of this note is to understand under which conditions invertible modules over a commutative S-algebra in the sense of Elmendorf, Kriz, Mandell and May give rise to elements in the algebraic Picard group of invertible graded modules…

We characterize the commutative rings whose ideals (resp. regular ideals) are products of radical ideals.

Let k be a field of arbitrary characteristic and let Q be a quiver of finite representation type. In this paper we prove that if M is an indecomposable kQ-module then the universal deformation ring of M over kQ is isomorphic to k.

A commutative ring R is locally stable provided that for any $a,b\in R$ such that $aR+bR=R$, there exist some $y\in R$ such that $R/(a+by)R$ has stable range 1.For a Bezout ring $R$, we prove that $R$ is an elementary divisor ring if and…

An $R$-module $M$ is called absolutely self pure if for any finitely generated left ideal of $R$ whose kernel is in the filter generated by the set of all left ideals $L$ of $R$ with $L \supseteq$ ann $(m)$ for some $m \in M$, any map from…

In this article, we study certain local cohomology modules over $F$-pure rings. We give sufficient conditions for the vanishing of some Lyubeznik numbers, derive a formula for computing these invariants when the $F$-pure ring is standard…

Let $R$ be a ring with unity. The clean graph $\text{Cl}(R)$ of a ring $R$ is the simple undirected graph whose vertices are of the form $(e,u)$, where $e$ is an idempotent element and $u$ is a unit of the ring $R$ and two vertices $(e,u)$,…

Let $(R, \m)$ be a commutative Noetherian local ring with $\m^3 =(0)$. We give a condition for $R$ to have a non-free module of G-dimension zero. We shall also construct a family of non-isomorphic indecomposable modules of G-dimension zero…

In this note, we establish conditions under which the union of an increasing sequence of completely decomposable modules over domains are again completely decomposable. In our investigation, the condition of purity of modules is crucial. In…

Let $R$ be a noetherian normal domain. We investigate when $R$ admits a faithful module whose endomorphism ring has finite global dimension. This can be viewed as a non-commutative desingularization of $\Spec(R)$. We show that the existence…