Related papers: Nil-good and nil-good clean matrix rings
In this paper, strongly clean ring defined by W. K. Nicholson in 1999 has been generalized to n-strongly clean, {\Sigma}-strongly clean and with the help of example it has been shown that there exists a ring, which is n-strongly clean and…
This paper presents an extension of the concept of NR-clean introduced in [12] to graded ring theory. We define and explore graded NR-clean rings, which generalize the class of graded U-nil clean previously studied in [15]. We provide…
A differential module is a module equipped with a square-zero endomorphism. This structure underpins complexes of modules over rings, as well as differential graded modules over graded rings. We establish lower bounds on the class--a…
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…
We introduce the class weakly nil clean rings, as rings R in which for every a\in R there exist an idempotent e and a nilpotent q such that a-e-q\in eRa. Every weakly nil clean ring is exchange. Weakly nil clean rings contain pi-regular…
We consider rings whose one-sided ideals are close to automorphism-invariant modules. We study rings in which every (finitely generated) right ideal is automorphism invariant and rings in which every right ideal is a finite direct sum of…
The main result is Theorem: Let A be an R-algebra, mu, lambda be cardinals such that |A|<=mu=mu^{aleph_0}<lambda<=2^mu. If A is aleph_0-cotorsion-free or A is countably free, respectively, then there exists an aleph_0-cotorsion-free or a…
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…
We prove that an integral Jacobson radical ring is always nil, which extends a well known result from algebras over fields to rings. As a consequence we show that if every element x of a ring R is a zero of some polynomial p_x with integer…
This paper investigates key properties of ZINC rings and their relationships with semicommutative and weakly semicommutative rings. We call an element $x$ of a ring $R$ zero insertive if $x=arb$ for some $a,b,r\in R$ such that $ab=0$ and…
We describe the endomorphism rings in an additive category whose objects are right $R$-modules $M$ with a fixed chain of submodules $0=M^{(0)}\leq M^{(1)}\leq M^{(2)} \leq \dots \leq M^{(n)}=M$ and the behaviour of these objects as far as…
In this article, we introduce the concept of nilpotent graph of a finite commutative ring. The set of all non nilpotent elements of a ring is taken as the vertex set and two vertices are adjacent if and only if their sum is nilpotent. We…
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.
We describe and present a new construction method for codes using encodings from group rings. They consist primarily of two types: zero-divisor and unit-derived codes. Previous codes from group rings focused on ideals; for example cyclic…
A ring with an involution * is called strongly $J$-*-clean if every element is a sum of a projection and an element of the Jacobson radical that commute. In this article, we prove several results characterizing this class of rings. It is…
Let R be a commutative ring with identity and N(R) be the set of all nilpotent elements of R. The aim of this paper is to introduce and study the notion of nil-prime ideals as a generalization of prime ideals. We say that a proper ideal P…
We define and consider in-depth the so-called $C\Delta$ rings as those rings $R$ whose elements are a sum of an element in $C(R)$ and of an element in $\Delta(R)$. Our achieved results somewhat strengthen these recently obtained by…
Various authors have been generalizing some unital ring properties to nonunital rings. We consider properties related to cancellation of modules (being unit-regular, having stable range one, being directly finite, exchange, or clean) and…
We survey theory developed over the past 10 years of semirings which need not be additively cancellative. The main feature is a specified ``null ideal'' $\mcA_0$ of a semiring $\mcA,$ taking the place of a zero element, which permits…
Motivated by Hirano-Tominaga's work \cite{HT} on rings for which every element is a sum of two idempotents and by de Seguins Pazzis's results \cite{de} on decomposing every matrix over a field of positive characteristic as a sum of…