Related papers: Nil Clean Involutions
We show that a ring $\,R\,$ has two idempotents $\,e,e'\,$ with an invertible commutator $\,ee'-e'e\,$ if and only if $\,R \cong {\mathbb M}_2(S)\,$ for a ring $\,S\,$ in which $\,1\,$ is a sum of two units. In this case, the…
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…
An equivalent condition for an element of a Lie algebra acting nilpotently in all its representations is obtained. Namely, it should belong to the derived algebra and go via factoring over the radical to a nilpotent element of the…
This article addresses two central problems in the theory of quandle rings. First, motivated by Conjecture 3.10 in Internat. J. Math. 34 (2023), no. 3, Paper No. 2350011: for a semi-latin quandle $X$, every nonzero idempotent in the…
In order to find a suitable expression of an arbitrary square matrix over an arbitrary finite commutative ring, we prove that every such a matrix is always representable as a sum of a potent matrix and a nilpotent matrix of order at most…
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 prove that every \omega-categorical, generically stable group is nilpotent-by-finite and that every \omega-categorical, generically stable ring is nilpotent-by-finite.
This paper, we consider some properties of rings via q-potent and periodic elements. In this paper we give some results of rings in which every element is a sum of an idempotent and a q-potent that commute; periodic rings and k-potent…
In this paper, we define and study a particular case of von Neumann regular notion called a weak von Neumann regular ring. It shown that the polynomial ring $R[x]$ is weak von Neumann regular if and only if $R$ has exactly two idempotent…
A well-known theorem of Wedderburn asserts that a finite division ring is commutative. In a division ring the group of invertible elements is as large as possible. Here we will be particularly interested in the case where this group is as…
Let $R$ be a finite ring with identity. The clean graph $Cl(R)$ of a ring $R$ is a graph whose vertices are pairs $(e, u)$, where $e$ is an idempotent element and $u$ is a unit of $R$. Two distinct vertices $(e, u)$ and $(f, v)$ are…
Hindman's Theorem says that every finite coloring of the positive natural numbers has a monochromatic set of finite sums. Ramsey algebras, recently introduced, are structures that satisfy an analogue of Hindman's Theorem. It is an open…
In this paper, we investigate idempotents in quandle rings and relate them with quandle coverings. We prove that integral quandle rings of quandles of finite type that are non-trivial coverings over nice base quandles admit infinitely many…
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 show that quandle rings and their idempotents lead to proper enhancements of the well-known quandle coloring invariant of links in the 3-space. We give explicit examples to show that the new invariants are also stronger than the $\Hom$…
In this paper, we investigate semirings whose elements are either units or zero-divisors (nilpotents) with many examples. While comparing these semirings with their counterparts in ring theory, we observe that their behavior is different in…
The type and several invariant subspaces related to the upper annihilating series of finite-dimensional nilpotent evolution algebras are introduced. These invariants can be easily computed from any natural basis. Some families of nilpotent…
We define the class of {\it CUSC} rings, that are those rings whose clean elements are uniquely strongly clean. These rings are a common generalization of the so-called {\it USC} rings, introduced by Chen-Wang-Zhou in J. Pure \& Applied…
Let $R$ be a ring, $e$ an idempotent of $R$ and $\delta(R)$ denote the intersection of all essential maximal right ideals of $R$ which is called Zhou radical. In this paper, the Zhou radical of a ring is applied to the $e$-reduced property…
A ring $R$ is periodic provided that for any $a\ in R$ there exist distinct elements $m,n \in {\Bbb N}$ such that $a^m=a^n$. We shall prove that periodicity is inherited by a type of generalized matrix rings.We define strongly periodic…