arXiv subjects

M. Sheibani

Publications and source records attributed to M. Sheibani.

4 recordsLinked to original sources

arXiv2025

Generalized weighted EP elements in Banach algebras

We propose a new class of generalized inverses with weights, which represent a natural extension of EP (Moore-Penrose) and *-DMP (Drazin-Moore-Penrose) elements in a Banach *-algebra. This paper presents various characteristics of weighted EP elements. Moreover, we characterize the weighted EP element through the core-EP decomposition and a polar-like property. Finally, we explore weighted *-DMP elements and uncover several new properties of *-DMP elements.

math.FA↗

arXiv2016

Rings over which every matrix is the sum of two idempotents and a nilpotent

A ring $R$ is (strongly) 2-nil-clean if every element in $R$ is the sum of two idempotents and a nilpotent (that commute). Fundamental properties of such rings are discussed. Let $R$ be a 2-primal ring. If $R$ is strongly 2-nil-clean, we show that $M_n(R)$ is 2-nil-clean for all $n\in {\Bbb N}$. We also prove that the matrix ring is 2-nil-clean for a strongly 2-nil-clean ring of bounded index. These provide many classes of rings over which every matrix is the sum of two idempotents and a nilpotent.

math.RA↗

arXiv2015

Uniquely weakly nil-clean conditions on zero-divisors

An element in a ring $R$ is called uniquely weakly nil-clean if every element in $R$ can be uniquely written as a sum or a difference of a nilpotent and an idempotent in the sense of very idempotents. The structure of the ring in which every zero-divisor is uniquely weakly nil-clean is completely determined. We prove that every zero-divisor in a ring $R$ is uniquely weakly nil-clean if and only if $R$ is a D-ring, or $R$ is abelian, periodic, and $R/J(R)$ is isomorphic to a field $F$, ${\Bbb Z}_{3}\oplus {\Bbb Z}_{3}$, ${\Bbb Z}_{3}\oplus B$ where $B$ is Boolean, or a Boolean ring. As a specific case, rings in which every zero-divisor $a$ or $-a$ is a nilpotent or an idempotent are also considered. Furthermore, we prove that every zero-divisor in a ring $R$ is uniquely nil-clean if and only if $R$ is a D-ring, or $R$ is abelian, periodic; and $R/J(R)$ is Boolean.\vskip3mm \no {\bf Key words}: Zero-divisor; Uniquely weakly nil-clean ring; Uniquely nil-clean ring.

math.RA↗

arXiv2015

Feckly Adequate Conditions and Elementary Matrix Reduction

We present some new conditions for a B$\acute{e}$zout ring to be an elementary divisor ring. We prove, in this note, that a B$\acute{e}$zout ring $R$ is feckly zero-adequate if and only if $R/J(R)$ is regular if and only if $R/J(R)$ is $π$-regular, and that every feckly zero-adequate ring is an elementary divisor ring. If $R$ has feckly adequate range 1, we prove that $R$ is an elementary divisor ring if and only if $R$ is a B$\acute{e}$zout ring. Many known results are thereby generalized to much wider class of rings, e.g. [4, Theorem 14], [5, Theorem 4], [8, Theorem 1.2.14], [10, Theorem 4] and [11, Theorem 7]. \vskip3mm {\bf Keywords:} Elementary divisor ring, B$\acute{e}$zout ring, Feckly zero-adequate ring, Feckly adequate range 1.

math.RA↗