Related papers: Matrix Algebras over Strongly Non-Singular Rings

We call a right module $M$ (strongly) virtually regular if every (finitely generated) cyclic submodule is isomorphic to a direct summand. $M$ is said to be completely virtually regular if every submodule is virtually regular. In this paper,…

Let $(R,\mathfrak{m},\mathbb{k})$ be an equicharacteristic one-dimensional complete local domain over an algebraically closed field $\mathbb{k}$ of characteristic 0. R. Berger conjectured that R is regular if and only if the universally…

Constructive proofs of fact that a stably free left $S$-module $M$ with rank$(M)\geq$sr$(S)$ is free, where sr$(S)$ denotes the stable rank of an arbitrary ring $S$, were developed in some articles. Additionally, in such papers, are…

We study rings with infinitely (only finitely) many maximal subrings. We prove that if $M$ is a maximal left/right ideal of a ring $T$ which is not an ideal of $T$, and $R$ is the idealizer of $M$, then $T$ has at least $|R/M|+1$ maximal…

A ring $R$ is (strongly) 2-nil-clean if every element in $R$ is the sum of two idempotents and a nilpotent (that commute). Fundamental properties of such rings are discussed. Let $R$ be a 2-primal ring. If $R$ is strongly 2-nil-clean, we…

A ring $R$ is uniquely (strongly) clean provided that for any $a\in R$ there exists a unique idempotent $e\in R \big(\in comm(a)\big)$ such that $a-e\in U(R)$. Let $R$ be a uniquely bleached ring. We prove, in this note, that $R$ is…

Following O'Meara's result [Journal of Algebra and Its Applications Vol~\textbf{13}, No. 8 (2014)], it follows that the block matrix $A=\begin{pmatrix} B & 0 0 & 0 \end{pmatrix} \in M_{n+r}(R)$, $B\in M_n(R)$, $r\ge 1$, over a von Neumann…

Let $(R, \mathfrak{m})$ be a commutative Noetherian local ring with total quotient ring $K$. An $R$-module $M$ is called simple divisible, if $M$ is divisible $\neq 0$, but every proper submodule $0 \neq U \subsetneqq M$ is not divisible.…

An element $x \in R$ is considered (strongly) nil-clean if it can be expressed as the sum of an idempotent $e \in R$ and a nilpotent $b \in R$ (where $eb = be$). If for any $x \in R$, there exists a unit $u \in R$ such that $ux$ is…

Let $\mathcal{P}$ be the class of rings for which every indecomposable right module is pure-projective or pure-injective. When $R$ is a Noetherian local commutative ring of maximal ideal $P$, it is proven that $R\in\mathcal{P}$ if and only…

This paper presents the right-left symmetry of the CS and max-min CS conditions on nonsingular rings, and generalization to nonsingular modules. We prove that a ring is right nonsingular right CS and left Utumi if and only if it is left…

In this paper we prove that two idempotent rings are Morita equivalent if every corner of one of them is isomorphic to a corner of a matrix ring of the other one. We establish the converse (which is not true in general) for $\sigma$-unital…

Many known results on finite von Neumann algebras are generalized, by purely algebraic proofs, to a certain class ${\mathcal C}$ of finite Baer *-rings. The results in this paper can also be viewed as a study of the properties of Baer…

We consider algebras in a modular tensor category C. If the trace pairing of an algebra A in C is non-degenerate we associate to A a commutative algebra Z(A), called the full centre, in a doubled version of the category C. We prove that two…

A monoid $S$ is right coherent if every finitely generated subact of every finitely presented right $S$-act is finitely presented. The corresponding notion for a ring $R$ states that every finitely generated submodule of every finitely…

We study the notion of Rickart property in a general module theoretic setting as a generalization to the concept of Baer modules and right Rickart rings. A module $M_{R}$ is called Rickart if the right annihilator in $M_{R}$ of each left…

In this paper, we introduce and study the class $S$-$\mathcal{F}$-ML of $S$-Mittag-Leffler modules with respect to all flat modules. We show that a ring $R$ is $S$-coherent if and only if $S$-$\mathcal{F}$-ML is closed under submodules. As…

We investigate the notion of \textit{semi-nil clean} rings, defined as those rings in which each element can be expressed as a sum of a periodic and a nilpotent element. Among our results, we show that if $R$ is a semi-nil clean NI ring,…

Let $(R,\mathfrak{m},k)$ be a Noetherian local ring and let $M$ be a finitely generated $R$-module. The main focus of this paper is to give positive answers for some long-standing homological conjectures over the idealization ring $R\ltimes…

Let $R$ be a commutative Noetherian local ring. Assume that $R$ has a pair $\{x,y\}$ of exact zerodivisors such that $\dim R/(x,y)\ge2$ and all totally reflexive $R/(x)$-modules are free. We show that the first and second Brauer--Thrall…