Related papers: Rings Whose Invertible Elements Are Weakly Nil-Cle…

In this paper, we study a new class of rings, called $\sqrt{J}$-clean rings. A ring in which every element can be expressed as the addition of an idempotent and an element from $\sqrt{J(R)}$ is called a $\sqrt{J}$-clean ring. Here,…

A ring R is a Zhou nil-clean ring if every element in R is the sum of two tripotents and a nilpotent that commute. In this paper, Zhou nil-clean rings are further discussed with an emphasis on their relations with polynomials, idempotents…

We introduce and study the so-called {\it weakly $\sqrt{J}U$ rings} (hereafter abbreviated as {\it $W\sqrt{J}U$ rings} for short), in which every unit is of the form $j+1$ or $j-1$ for some $j$ in $\sqrt{J(R)} : = \{x \in R : x^n \in J(R)…

Let $R$ be a commutative ring with identity. An element $r \in R$ is said to be absolutely irreducible in $R$ if for all natural numbers $n>1$, $r^n$ has essentially only one factorization namely $r^n = r \cdots r$. If $r \in R$ is…

We examine those matrix rings whose entries lie in periodic rings equipped with some additional properties. Specifically, we prove that the famous Diesl's question whether or not $R$ being nil-clean implies that $\mathbb{M}_n(R)$ is…

We introduce and study a nontrivial generalization of uniserial modules and rings. A module is called weakly uniserial if its submodules are comparable regarding embedding. Also, a right (resp., left) weakly uniserial ring is a ring which…

Let $R$ be an associative ring with identity, $C(R)$ denote the center of $R$, and $g(x)$ be a polynomial in the polynomial ring $C(R)[x]$. $R$ is called strongly $g(x)$-clean if every element $r \in R$ can be written as $r=s+u$ with…

We study the class of rings $R$ with the property that for $x\in R$ at least one of the elements $x$ and $1+x$ are tripotent.

A commutative ring $R$ is projective free provided that every finitely generated $R$-module is free. An element in a ring is strongly clean provided that it is the sum of an idempotent and a unit that commutates. Let $R$ be a…

Let R be a commutative ring with $1\neq0$. In this paper, we introduce the concept of weakly 1-absorbing primary ideal which is a generalization of 1-absorbing ideal. A proper ideal $I$ of $R$ is called a weakly 1-absorbing primary ideal if…

An ideal $I$ of a ring $R$ is called left N-reflexive if for any $a\in$ nil$(R)$, $b\in R$, being $aRb \subseteq I$ implies $bRa \subseteq I$ where nil$(R)$ is the set of all nilpotent elements of $R$. The ring $R$ is called left…

Let R be a ring with the set of nilpotents Nil(R). We prove that the following are equivalent: (i) Nil(R) is additively closed, (ii) Nil(R) is multiplicatively closed and R satisfies Koethe's conjecture, (iii) Nil(R) is closed under the…

A ring $R$ is said to be i-reversible if for every $a,b$ $\in$ $R$, $ab$ is a non-zero idempotent implies $ba$ is an idempotent. It is known that the rings $M_n(R)$ and $T_n(R)$ (the ring of all upper triangular matrices over $R$) are not…

An element $a$ of a ring $R$ is called \emph{strongly $J$-clean} provided that there exists an idempotent $e\in R$ such that $a-e\in J(R)$ and $ae=ea$. A ring $R$ is \emph{strongly $J$-clean} in case every element in $R$ is strongly…

This paper introduces and studies a new class of rings called {\it $U\sqrt{\Delta}$-rings}. A ring $R$ is $U\sqrt{\Delta}$ if every non-unit element can be written as the product of a unit and an element from $\sqrt{\Delta(R)}$, where…

An element in a ring $R$ is called clear if it is the sum of unit-regular element and unit. An associative ring is clear if every its element is clear. In this paper we defined clear rings and extended many results to wider class. Finally,…

An element of a ring is unique clean if it can be uniquely written as the sum of an idempotent and a unit. A ring $R$ is uniquely $\pi$-clean if some power of every element in $R$ is uniquely clean. In this article, we prove that a ring $R$…

Let $R$ be a commutative local ring. It is proved that $R$ is Henselian if and only if each $R$-algebra which is a direct limit of module finite $R$-algebras is strongly clean. So, the matrix ring $\mathbb{M}_n(R)$ is strongly clean for…

We call a ring $R$ NJ-symmetric if $abc\in N(R)$ implies $bac\in J(R)$ for any $a,b,c\in R$. Some classes of rings that are NJ-symmetric include left (right) quasi-duo rings, weak symmetric rings, and abelian J-clean rings. We observe that…

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…