Related papers: Pointwise Semicommutative Rings

In this paper, we introduce a class of $J$-quasipolar rings. Let $R$ be a ring with identity. An element $a$ of a ring $R$ is called {\it weakly $J$-quasipolar} if there exists $p^2 = p\in comm^2(a)$ such that $a + p$ or $a-p$ are contained…

In this paper all rings are commutative. We prove some new results on flat epimorphisms of rings and pointwise localizations. Especially among them, it is proved that a ring $R$ is an absolutely flat (von-Neumann regular) ring if and only…

Let $R$ be a commutative ring with $ 1 \neq 0$. We recall that a proper ideal $I$ of $R$ is called a semiprimary ideal of $R$ if whenever $a,b\in R$ and $ab \in I$, then $a\in \sqrt{I}$ or $b\in \sqrt{I}$. We say $I$ is a {\it weakly…

An element $a$ of a ring $R$ is called perfectly clean if there exists an idempotent $e\in comm^2(a)$ such that $a-e\in U(R)$. A ring $R$ is perfectly clean in case every element in $R$ is perfectly clean. In this paper, we investigate…

Regarding the question of how idempotent elements affect reversible property of rings, we study a version of reversibility depending on idempotents. In this perspective, we introduce {\it right} (resp., {\it left}) {\it $e$-reversible…

Let $\mathscr{R}$ be a prime ring of Char$(\mathscr{R}) \neq 2$ and $m\neq 1$ be a positive integer. If $S$ is a nonzero skew derivation with an associated automorphism $\mathscr{T}$ of $\mathscr{R}$ such that $([S([a, b]), [a, b]])^{m} =…

'A semigroup is completely regular if and only if it is a union of groups'- an analogue of this structure theorem of completely regular semigroup has been obtained in the setting of seminearrings in [[16], Mukherjee (Pal) et al., Semigroup…

An element $a$ of a ring $R$ is called \emph{quasipolar} provided that there exists an idempotent $p\in R$ such that $p\in comm^2(a)$, $a+p\in U(R)$ and $ap\in R^{qnil}$. A ring $R$ is \emph{quasipolar} in case every element in $R$ is…

A new class of rings, {\em the class of weakly left localizable rings}, is introduced. A ring $R$ is called {\em weakly left localizable} if each non-nilpotent element of $R$ is invertible in some left localization $S^{-1}R$ of the ring…

An element of a ring $R$ is strongly $P$-clean provided that it can be written as the sum of an idempotent and a strongly nilpotent element that commute. A ring $R$ is strongly $P$-clean in case each of its elements is strongly $P$-clean.…

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.

Let R be a commutative ring with unity $1\in R$. In this article, we introduce the concept of prime principal right ideal rings (\textbf{PPRIR}), A prime ideal P of R is said to be prime principal right ideal (\textbf{PPRI}) is given by $P…

In this paper, we introduce a class of quasipolar rings which is a generalization of $J$-quasipolar rings. Let $R$ be a ring with identity. An element $a \in R$ is called {\it $\delta$-quasipolar} if there exists $p^2 = p\in comm^2(a)$ such…

Irreducible decompositions of monomial ideals in polynomial rings over a field are well-understood. In this paper, we investigate decompositions in the set of monomial ideals in the semigroup ring A[\mathbb{R}_{\geq 0}^d] where A is an…

In this paper, we describe finite, additively commutative, congruence simple semirings. The main result is that the only such semirings are those of order 2, zero-multiplication rings of prime order, matrix rings over finite fields, those…

Let $S$ be a commutative ring with identity and $R$ a unitary subring of $S$. An ideal $I$ of $S$ is called an $R$-conductor ideal of $S$ if $I=\{x\in S\mid xS\subseteq V\}$ for some intermediate ring $V$ of $R$ and $S$. In this note we…

The purpose of this article is to define and examine graded almost prime ideals over a non-commutative graded ring, and consider some cases where all graded right ideals of a non-commutative graded ring are graded almost prime.

In this paper we prove that if $R$ is a commutative refinement ring and $M$, $N$ are two $R$-modules then, $M\cong N$ if and only if for every maximal ideal $m$ of $R$, $M_m\cong N_m$. We prove if $R$ is a refinement ring, then every…

Let $RG$ denote the group ring of the torsion group $G$ over a commutative ring $R$ with identity. In this paper we present proofs of some statements that appear without to be proved in the literature. We establish the valid implications…

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…