Related papers: Weakly $I$-clean rings

We study those rings in which all invertible elements are weakly nil-clean calling them {\it UWNC rings}. This somewhat extends results due to Karimi-Mansoub et al. in Contemp. Math. (2018), where rings in which all invertible elements are…

A *-ring $R$ is called a strongly nil-*-clean ring if every element of $R$ is the sum of a projection and a nilpotent element that commute with each other. In this article, we show that $R$ is a strongly nil-*-clean ring if and only if…

The target of the present work is to give a new insight in the theory of {\it strongly weakly nil-clean} rings, recently defined by Kosan and Zhou in the Front. Math. China (2016) and further explored in detail by Chen-Sheibani in the J.…

A $*$-ring $R$ is called (strongly) $*$-clean if every element of $R$ is the sum of a projection and a unit (which commute with each other). In this note, some properties of $*$-clean rings are considered. In particular, a new class of…

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 is clean (almost clean) if each of its elements is the sum of a unit (regular element) and an idempotent. A module is clean (almost clean) if its endomorphism ring is clean (almost clean). We show that every quasi-continuous and…

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.…

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…

An element of a ring R is called clean if it is the sum of an idempotent and a unit. A ring R is called clean if each of its element is clean. An element r \in R called regular if r = ryr for some y \in R. The ring R is regular if each of…

In this note we show that a ring R is left perfect if and only if every left R-module is weakly supplemented if and only if R is semilocal and the radical of the countably infinite free left R-module has a weak supplement.

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…

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…

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,…

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

Many authors have investigated the behavior of strong cleanness under certain ring extensions. In this note, we prove that if $R$ is a ring which is complete with respect to an ideal $I$ and if $x$ is an element of $R$ whose image in $R/I$…

A ring $R$ is feckly clean provided that for any $a\in R$ there exists an element $e\in R$ and a full element $u\in R$ such that $a=e+u, eR(1-e)\subseteq J(R)$. We prove that a ring $R$ is feckly clean if and only if for any $a\in R$, there…

We define here the notion of a {\it weakly reversible ring} $R$ saying that a non-zero element $a\in R$ is weakly reversible if there exists an integer $m>0$ depending on $a$ such that $a^m\neq 0$ is reversible, that is,…

Let R be a ring and let G be a group. We prove a rather curious necessary and sufficient condition for the commutative group ring RG to be weakly nil-neat only in terms of R,G and their sections. This somewhat expands three recent results,…

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,…