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Mehrdad Esfandiar

Publications and source records attributed to Mehrdad Esfandiar.

4 recordsLinked to original sources
arXiv2026

Rings Whose Non-Invertible Elements Are Uniquely Strongly Clean

We define and investigate in details the class of so-termed {\it GUSC} rings, that are those rings whose non-invertible elements are uniquely strongly clean. These rings are a common non-trivial generalization of the so-called {\it USC} rings, introduced by Chen-Wang-Zhou in J. Pure \& Appl. Algebra (2009), which are rings whose elements are uniquely strongly clean. These rings also properly generalize the so-named {\it GUC} rings, defined by Guo-Jiang in Bull. Transilvania Univ. Braşov (2023), which are rings whose non-invertible elements are uniquely clean.

math.RA↗
arXiv2026

A Generalization of $Δ$U Rings

In this paper, we introduce and study a new class of rings calling them {\it weakly $ΔU$-rings}, hereafter abbreviated as {\it $WΔU$-rings} for short. A ring $R$ is said to be $WΔU$ if every unit of $R$ can be expressed as $\pm 1 + d$ for some $d \in Δ(R)$, where $Δ(R)$ is the largest Jacobson radical of $R$ that is closed under multiplication by units. Utilizing the known structure of $Δ(R)$, we investigate the relationships between $WΔU$ rings and certain classical concepts such as $ΔU$-rings, $UJ$-rings, $WUJ$-rings, as well as clean and exchange rings. Among the main results, we show that a matrix ring $M_n(R)$ is never $WΔU$ for any $n \ge 2$. We also provide complete characterizations of local, semi-local, semi-simple and semi-regular rings that are $WΔU$. Furthermore, it is shown for exchange rings that the $WΔU$ property is equivalent to being $WUJ$. Furthermore, the behavior of $WΔU$-rings under various ring extensions, including skew polynomial rings, skew power series rings, triangular matrix rings, trivial extensions and group rings, is thoroughly examined. Several examples are given to illustrate that the class of $WΔU$-rings properly contains the class of $ΔU$-rings. Finally, necessary and sufficient conditions for a group ring $RG$ to be $WΔU$ are established too. Resuming all of the presented above, our results expanded those by Karabaçak et al. published in J. Algebra \& Appl. (2021).

math.RA↗
arXiv2025

Rings Such That $u-1$ Lies In $J^{\#}(R)$ For Each Unit $u$

We investigate the so-called {\it $UJ^{\#}$ rings}, a new type of rings in which every unit can be written as $1+j$ with $j\in J^{\#}(R)$. These rings were defined and studied by Saini-Udar in Czechoslovak Math. J. (2025) under the name {\it $\sqrt{J}U$ rings}. (See \cite{SU}.) This class extends both the classes of UU and UJ rings, but also has its own special properties. In this study, we present some additional results about $UJ^{\#}$ rings that supply those from \cite{SU} explaining their connections with Dedekind-finite, semi-potent and Boolean rings, respectively, as well as we give several characterizations in this direction. We also examine how these rings behave under common ring constructions and find conditions for group rings to be $UJ^{\#}$. Moreover, our establishments shed a clearer picture of how unit elements interact with radical-like parts of a ring.

math.RA↗
arXiv2025

On Strongly \( J^{\#} \)-Clean Rings

We define and examine the class of {\it strongly \( J^{\#} \)-clean rings} consisting of those rings $R$ such that each element of $R$ is the sum of an idempotent from $R$ and an element from $J^{\#}(R)$ that commute with each other. More exactly, we prove that these rings are simultaneously strongly clean and Dedekind-finite as well as that they factor-ring modulo the Jacobson radical is always Boolean, and also provide some close relations with certain other well-established classes of rings like these of local, semi-local and strongly J-clean rings (as introduced by Chen on 2010) showing the surprising fact that the classes of strongly \( J^{\#} \)-clean and strongly J-clean rings, actually, do coincide. Moreover, a few more extensions of the newly defined class such as group rings and generalized matrix rings are provided too.

math.RA↗