Related papers: Rings Whose Non-Invertible Elements Are Nil-Clean
We characterize the nil clean matrix rings over fields. As a by product, it is proved that the full matrix rings with coefficients in commutative nil-clean rings are nil-clean, and we obtain a complete characterization of the finite rank…
The behavior of nilpotents can reveal valuable information about the algebra. We give a simple proof of a classic result that a finite ring is commutative if all its nilpotents lie in the center.
In this paper, we are mainly interested in the two questions "which are the commutative rings on which every finitely presented modules is [Formula: see text]-periodic (respectively, [Formula: see text]-periodic)?". It is proved that these…
In recent years, the finite W-algebras associated to a semisimple Lie algebra and its nilpotent element have been studied intensively from different viewpoints. In this lecture series, we shall present some basic constructions, connections,…
Suppose that $G$ is a finite group and $k$ is a field of characteristic $p>0$. We consider the complete cohomology ring $\mathcal{E}_M^* = \sum_{n \in \mathbb{Z}} \widehat{Ext}^n_{kG}(M,M)$. We show that the ring has two distinguished…
A relationship between nilpotency and primeness in a module is investigated. Reduced modules are expressed as sums of prime modules. It is shown that presence of nilpotent module elements inhibits a module from possessing good structural…
In this article, the concept of nil clean graph of a ring has been generalised to weakly nil clean graph of a ring and graph theoretic properties like girth, clique number, diameter and chromatic index of the graph have been studied for a…
The purpose of this work is to extend the study of the commutative rings whose lattice of ideals can be a structure of BL-algebra as carry out by Heubo et al in 2018, to non commutative rings appointed in the work as pseudo BL-rings. We…
Inner relations are derived between partial augmentations of certain elements (units or idempotents) in group rings.
In this paper, we study the classes of rings in which every proper (regular) ideal can be factored as an invertible ideal times a nonempty product of proper radical ideals. More precisely, we investigate the stability of these properties…
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…
It is known that any torsion element in a lambda-ring is nilpotent. In this note we deduce a sharp estimate for the nilpotence degree of such an element.
In a paper on the taxonomy of 2-primal rings, examples of various types of rings that are related to commutativity such as reduced, symmetric, duo, reversible and PS~I were given in order to show that the ring class inclusions were strict.…
We provide the structure of some idempotents in the integral ring of a dihedral quandles, for odd case. The solution is complete in case $n=5$. This produces an example of a connected quandle $X$ such that $Aut (X)\not\cong Aut…
We first establish a sufficient condition for an additively idempotent semiring to be nonfinitely based. As applications, we exhibit several examples of additively idempotent semirings satisfying this condition, including a $4$-element…
In this paper, we say a ring $R$ is Nil$_{\ast}$-Noetherian provided that any nil ideal is finitely generated. First, we show that the Hilbert basis theorem holds for Nil$_{\ast}$-Noetherian rings, that is, $R$ is Nil$_{\ast}$-Noetherian if…
In this paper, we investigate zero-divisor, nilpotent, idempotent, unit, small, and irreducible elements in semiring extensions such as amount, content, and monoid semialgebras. We also introduce new concepts such as the prime avoidance…
The three famous problems concerning units, zero-divisors and idempotents in group rings of torsion-free groups, commonly attributed to I. Kaplansky, have been around for more than 60 years and still remain open in characteristic zero. In…
Since for the classification of finite (congruence-)simple semirings it remains to classify the additively idempotent semirings, we progress on the characterization of finite simple additively idempotent semirings as semirings of…
In this paper, we consider the N-pure notion. An ideal $I$ of a ring $R$ is said to be N-pure, if for every $a\in I$ there exists $b\in I$ such that $a(1-b)\in N(R)$, where N(R) is nil radical of $R$. We provide new characterizations for…