arXiv subjects

Mina Doostalizadeh

Publications and source records attributed to Mina Doostalizadeh.

7 recordsLinked to original sources

arXiv2026

A Generalization of UQ Rings

We examine the newly defined class of {\it $n$-$UQ$ rings} described by the condition that $u^n - 1 \in QN(R)$ for every unit $u \in U(R)$, where $QN(R)$ denotes the set of quasi-nilpotent elements (see \cite{Tien}). This class naturally extends the recently defined class of rings in \cite{daoa} and \cite{dam}, as well as expectedly generalizes previously explored concepts such as $UJ$, $UU$ and $UQ$ rings. We conduct here a comprehensive structural analysis of these $n$-$UQ$ rings and study their stability under various ring-theoretic constructions including matrix rings, group rings, trivial extensions and power series rings. As a result, several new characterizations are established, thus revealing relevant connections between $n$-$UQ$ rings and fundamental classes of rings such as reduced, clean, exchange, semi-regular and potent rings, respectively. Moreover, we prove that the classes of $n$-$UJ$ and $n$-$UU$ rings are properly contained in the class of $n$-$UQ$ rings. These achievements not only unify and expand existing theories in this branch, but also provide a robust framework for possible further investigations into the interplay between the unit behavior and quasi-nilpotency in noncommutative ring theory.

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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.

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arXiv2025

Unit Uniquely Clean Rings

We define the class of {\it unit uniquely clean} rings ({\it UnitUC} for short), that is a common generalization of uniquely clean rings and strongly nil clean rings. Abelian {\it UnitUC} rings are uniquely clean and {\it UnitUC} rings with nil Jacobson radical are strongly nil clean. These rings also generalize the UUC and CUC rings, defined by Calugareanu-Zhou in Mediterranean J. Math. (2023), which are rings whose clean elements are uniquely clean. These rings are also represent a natural generalization of the Boolian rings in that a ring is {\it UnitUC} if, and only if, it is exchange and Boolean modulo the Jacobson radical. The behavior of {\it UnitUC} rings under group ring and matrix ring extensions is investigated. Several examples are provided to explain and delimit the results.

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arXiv2025

Weakly Strongly 2-Nil-Clean Rings

In this paper, we introduce and explore in-depth the notion of {\it weakly strongly 2-nil-clean rings} as a common non-trivial generalization of both strongly 2-nil-clean rings and strongly weakly nil-clean rings as defined and studied by Chen-Sheibani in the J. Algebra \& Appl. (2017). We, specifically, succeeded to prove that any weakly strongly 2-nil-clean ring is strongly $π$-regular and, concretely, it decomposes as the direct product of a strongly 2-nil-clean ring and a ring of the type $\mathbb{Z}_{2^k}$ for some $k\geq 1$.

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arXiv2025

Rings Whose Non-Units are Square-Nil Clean

We consider in-depth and characterize in certain aspects the class of so-called {\it strongly NUS-nil clean rings}, that are those rings whose non-units are {\it square nil-clean} in the sense that they are a sum of a nilpotent and a square-idempotent that commutes with each other. This class of rings lies properly between the classes of strongly nil-clean rings and strongly clean rings. In fact, it is proved the valuable criterion that a ring $R$ is strongly NUS-nil clean if, and only if, $a^4-a^2\in Nil(R)$ for every $a\not\in U(R)$. In particular, a ring $R$ with only trivial idempotents is strongly NUS-nil clean if, and only if, $R$ is a local ring with nil Jacobson radical. Some special matrix constructions and group ring extensions will provide us with new sources of examples of NUS-nil clean rings.

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arXiv2025

Rings Whose Non-Invertible Elements are Strongly Weakly Nil-Clean

The target of the present work is to give a new insight in the theory of {\it strongly weakly nil-clean} rings, recently defined by Kosan and Zhou in the Front. Math. China (2016) and further explored in detail by Chen-Sheibani in the J. Algebra Appl. (2017). Indeed, we consider those rings whose non-units are strongly weakly nil-clean and succeed to establish that this class of rings is strongly $π$-regular and, even something more, that it possesses a complete characterization in terms of the Jacobson radical and sections of the $2\times 2$ full matrix ring. Additionally, some extensions like Morita context rings and groups rings are also studied in this directory.

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arXiv2024

Rings such that, for each unit $u$, $u^n-1$ belongs to the $Δ(R)$

We study in-depth those rings $R$ for which, there exists a fixed $n\geq 1$, such that $u^n-1$ lies in the subring $Δ(R)$ of $R$ for every unit $u\in R$. We succeeded to describe for any $n\geq 1$ all reduced $π$-regular $(2n-1)$-$Δ$U rings by showing that they satisfy the equation $x^{2n}=x$ as well as to prove that the property of being exchange and clean are tantamount in the class of $(2n-1)$-$Δ$U rings. These achievements considerably extend results established by Danchev (Rend. Sem. Mat. Univ. Pol. Torino, 2019) and Koşan et al. (Hacettepe J. Math. \& Stat., 2020). Some other closely related results of this branch are also established.

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