arXiv subjects

Adel N. Abyzov

Publications and source records attributed to Adel N. Abyzov.

3 recordsLinked to original sources

arXiv2026

Rings with Clean-Like Properties: Endomorphism, Matrix and Structural Theorems

We investigate three clean-like properties for arbitrary rings, for endomorphism rings of abelian groups and for matrix rings over finite fields. Specifically, we study and establish when a ring is weakly strongly $k$-nil-clean for some fixed natural number $k\geq 2$, when the matrix ring is either quasi $2$-nil-clean or quasi $3$-nil-clean, and when the endomorphism ring is weakly clean. Our theorems improve substantially on some results due to Goldsmith-Vámos in Rend. Sem. Mat. Univ. Padova (2007), Breaz {\it et al}. in Linear Algebra \& Appl. (2013), Koşan-Zhou in Front. Math. China (2016), Su {\it et al}. in J. Algebra \& Appl. (2027), and some other existing results in this current topic.

math.RA↗

arXiv2023

Rings and finite fields whose elements are sums or differences of tripotents and potents

We significantly strengthen results on the structure of matrix rings over finite fields and apply them to describe the structure of the so-called weakly $n$-torsion clean rings. Specifically, we establish that, for any field $F$ with either exactly seven or strictly more than nine elements, each matrix over $F$ is presentable as a sum of of a tripotent matrix and a $q$-potent matrix if and only if each element in $F$ is presentable as a sum of a tripotent and a $q$-potent, whenever $q>1$ is an odd integer. In addition, if $Q$ is a power of an odd prime and $F$ is a field of odd characteristic, having cardinality strictly greater than $9$, then, for all $n\geq 1$, the matrix ring $\mathbb{M}_n(F)$ is weakly $(Q-1)$-torsion clean if and only if $F$ is a finite field of cardinality $Q$. A novel contribution to the ring-theoretical theme of this study is the classification of finite fields $\FQ$ of odd order in which every element is the sum of a tripotent and a potent. In this regard, we obtain an expression for the number of consecutive triples $γ-1,γ,γ+1$ of non-square elements in $\FQ$; in particular, $\FQ$ contains three consecutive non-square elements whenever $\FQ$ contains more than 9 elements.

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arXiv2023

Rings close to periodic with applications to matrix, endomorphism and group rings

We examine those matrix rings whose entries lie in periodic rings equipped with some additional properties. Specifically, we prove that the famous Diesl's question whether or not $R$ being nil-clean implies that $\mathbb{M}_n(R)$ is nil-clean for all $n\geq 1$ is paralleling to the corresponding implication for (Abelian, local) periodic rings. Besides, we study when the endomorphism ring $\mathrm{E}(G)$ of an Abelian group $G$ is periodic. Concretely, we establish that $\mathrm{E}(G)$ is periodic exactly when $G$ is finite as well as we find a complete necessary and sufficient condition when the endomorphism ring over an Abelian group is strongly $m$-nil clean for some natural number $m$ thus refining an "old" result concerning strongly nil-clean endomorphism rings. Responding to a question when a group ring is periodic, we show that if $R$ is a right (resp., left) perfect periodic ring and $G$ is a locally finite group, then the group ring $RG$ is periodic, too. We finally find some criteria under certain conditions when the tensor product of two periodic algebras over a commutative ring is again periodic. In addition, some other sorts of rings very close to periodic rings, namely the so-called weakly periodic rings, are also investigated.

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