Related papers: On graded nil clean rings

An element $a$ in a ring $R$ is strongly J-clean if it is the sum of an idempotent and an element in the Jacobson radical that commutes. We characterize the strongly J-clean $2\times 2$ matrices over 2-projective-free non-commutative rings.

In this paper we consider the question of when the associated graded ring along a valuation, ${\rm gr}_{\nu^*}(S)$, is a finite ${\rm gr}_{\nu^*}(R)$-module, where $S$ is a normal local ring which lies over a normal local ring $R$ and…

We show that if a groupoid graded ring has a certain nonzero ideal property and the principal component of the ring is commutative, then the intersection of a nonzero twosided ideal of the ring with the commutant of the principal component…

We study modular forms of some congruence subgroups. In this paper, we treat the cases level is 2-power, 3-power or 5. Structures of graded rings and many identities of infinite sum or infinite product are given. Theory of rational (1/3,…

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,…

The current paper is dedicated to the study of the classical $K_1$ groups of graded rings. Let $A$ be a $\Gamma$ graded ring with identity $1$, where the grading $\Gamma$ is an abelian group. We associate a category with suspension to the…

In this article we prove several important results on graded rings, especially monoid-rings, that are motivated and inspired by Kaplansky's zero-divisor, unit and idempotents conjectures. Among the main results, we first generalize…

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…

Like the lower central series of a nilpotent group, filters generalize the connection between nilpotent groups and graded Lie rings. However, unlike the case with the lower central series, the associated graded Lie ring may share few…

Let $G$ be a group with identity $e$ and $R$ be a $G$-graded commutative ring with nonzero unity $1$. In this article, we introduce the concept of graded strongly $1$-absorbing primary ideals. A proper graded ideal $P$ of $R$ is said to be…

The mod-p cohomology ring of a non-trivial finite p-group is an infinite dimensional, finitely presented graded unital algebra over the field with p elements, with generators in positive degrees. We describe an effective algorithm to test…

An order is a commutative ring that as an abelian group is finitely generated and free. A commutative ring is reduced if it has no non-zero nilpotent elements. In this paper we use a new tool, namely, the fact that every reduced order has a…

We investigate the graded Brown--McCoy and the classical Brown--McCoy radical of a graded ring, which is the direct sum of a family of its additive subgroups indexed by a nonempty set, under the assumption that the product of homogeneous…

This monograph is devoted to a comprehensive study of graded rings and graded K-theory. A bird's eye view of the graded module theory over a graded ring gives an impression of the module theory with the added adjective "graded" to all its…

The goal of this article is to propose and examine the notion of graded classical weakly prime submodules over non-commutative graded rings which is a generalization of the concept of graded classical weakly prime submodules over…

We introduce an analogue to Quasi-$F$-splittings, Quasi-$F$-purity, which is definable over rings that are not necessarily $F$-finite. We show that this property is equivalent to being Quasi-$F$-split in the complete local and $F$-finite…

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.

A ring $R$ is nil-clean if every element in $R$ is the sum of an idempotent and a nilpotent. A ring $R$ is abelian if every idempotent is central. We prove that if $R$ is abelian then $M_n(R)$ is nil-clean if and only if $R/J(R)$ is Boolean…

In this paper, we study the structure of the graded ring associated to a limit key polynomial $Q_n$ in terms of the key polynomials that define $Q_n$. In order to do that, we use direct limits. In general, we describe the direct limit of a…

In this paper we show that for a torsion-free abelian group $G$, $\operatorname{rank}_\mathbb{Z}G<\infty$ if and only if there exists a Noetherian $G$-graded ring $R$ such that the set $\{R_g \neq 0\}$ generates the group $G$. For every $G$…