Related papers: On Perfectly Clean Rings

This work is a review of results about centrally essential rings and semirings. A ring (resp., semiring) is said to be centrally essential if it is either commutative or satisfy the property that for any non-central element $a$, there exist…

In this paper we discuss several constructions that lead to new examples of nil-clean, clean, and exchange rings. A characterization of the idempotents in the algebra defined by a 2-cocycle is given and used to prove some of the algebra's…

Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. In this paper, we first introduce and study the notions of $s$-pure exact sequences and $s$-absolutely pure modules which extend the classical notions of…

We describe isomorphisms between strongly triangular matrix rings that were defined earlier in Berkenmeier et al. (2000) as ones having a complete set of triangulating idempotents, and we show that the so-called triangulating idempotents…

Let $R$ be a finite ring with identity. The idempotent graph $I(R)$ is the graph whose vertex set consists of the non-trivial idempotent elements of $R$, where two distinct vertices $x$ and $y$ are adjacent if and only if $xy = yx = 0$. The…

In 2011, Khurana, Lam and Wang define the following property. (*)A commutative unital ring A satisfies the property ''power stable range one'' if for all a, b $\in$ A with aA + bA = A there are an integer N = N (a, b) $\ge$ 1 and $\lambda$…

For a nonempty subset $X$ of a ring $R$, the ring $R$ is called $X$-semiprime if, given $a\in R$, $aXa=0$ implies $a=0$. This provides a proper class of semiprime rings. First, we clarify the relationship between idempotent semiprime and…

In this article, we introduce a new graph theoretic structure associated with a finite commutative ring, called nil clean divisor graph. For a ring $R$, nil clean divisor graph is denoted by $G_N(R)$, where the vertex set is $\{x\in R\,:\,…

We study clean group rings and also the group rings whose every element is a sum of two units. We also prove that if R is an Abelian exchange ring and G is a locally finite group, then the group ring RG has stable range one.

Let $a$ be a regular element of a ring $R$. If either $K:=\rm{r}_R(a)$ has the exchange property or every power of $a$ is regular, then we prove that for every positive integer $n$ there exist decompositions $$ R_R = K \oplus X_n \oplus Y_n…

Let $I$ denote an ideal in a commutative Noetherian ring $R$. Let $M$ be an $R$-module. The $I$-adic completion is defined by $\hat{M}^I = \varprojlim{}_{\alpha} M/I^{\alpha}M$. Then $M$ is called $I$-adic complete whenever the natural…

We point out that the usual argument used to prove that $R$ is strongly $F$-regular if and only if $R_{Q}$ is strongly $F$-regular for every prime ideal $Q \in \Spec R$, does not generalize to the case of pairs $(R, \ba^t)$. The author's…

We show that a unital ring is generated by its commutators as an ideal if and only if there exists a natural number $N$ such that every element is a sum of $N$ products of pairs of commutators. We show that one can take $N \leq 2$ for…

This article deals mostly with the following question: when is the classical ring of quotients of a commutative ring a ring of stable range 1? We introduce the concepts of a ring of (von Neumann) regular range 1, a ring of semihereditary…

We study integrality over rings (all commutative in this paper) and over ideal semifiltrations (a generalization of integrality over ideals). We begin by reproving classical results, such as a version of the "faithful module" criterion for…

In regard to our recent studies of rings with (strongly, weakly) nil-clean-like properties, we explore in-depth both the structural and characterization properties of those rings whose elements that are not units are weakly nil-clean. Group…

In this paper, we introduce and explore in-depth the notion of {\it weakly strongly 2-nil-clean rings} as a common non-trivial generalization of both strongly 2-nil-clean rings and strongly weakly nil-clean rings as defined and studied by…

Let $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be monomial ideal of $R$. In this paper, we show that if $I$ is a generic monomial ideal, then $R/I$ is pretty clean if and only if $R/I$ is…

Through this paper, we study the rings in which every unit's square is an element of the set $1+\sqrt{J(R)}$, and call them $2-\sqrt{J}U$ rings. Here, $\sqrt{J(R)}=\{x \in R: x^m \in J(R)$ for some $m \geq 1 \}$. We show that every…

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…