arXiv subjects

Ahmad Moussavi

Publications and source records attributed to Ahmad Moussavi.

At least 19 recordsLinked to original sources

arXiv2026

$\sqrtΔ$-Fine Rings

We introduce and study the so-termed {\it $\sqrtΔ$-fine rings}, a new class of rings that generalizes the classical {\it fine rings} introduced by Călugăreanu-Lam in J. Algebra \& Appl. (2016) by requiring that every nonzero element $r \in R$ can be written as $r = u + a$, where $u$ is a unit and $a \in \sqrt{Δ(R)}$. We establish that every such ring is simple, every abelian $\sqrtΔ$-fine ring is indecomposable, and most notably, the matrix ring $M_n(R)$ over a $\sqrtΔ$-fine ring $R$ is again $\sqrtΔ$-fine for every $n \ge 1$. As a consequence, we characterize all semi-local $\sqrtΔ$-fine rings as those rings which are precisely the simple Artinian rings. We also examine group rings, providing conditions under which they are either $\sqrtΔ$-fine or generalized fine, where the latter class was introduced by Zhou in J. Algebra \& Appl. (2022), and conclude our work with the difficult open question asking of whether each $\sqrtΔ$-fine ring is necessarily fine.

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arXiv2026

Expanding Generalized Fine Rings

We introduce and study the so-called {\it generalized $\sqrt{J}$-fine rings}, where every element outside the Jacobson radical is the sum of a unit and an element from the set $\sqrt{J(R)} := \{ x \in R : x^{n} \in J(R) \text{ for some } n \ge 1 \}$. This commonly extends the notions of {\it fine} and {\it generalized fine rings} defined, respectively, by Călugăreanu-Lam (J. Algebra \& Appl., 2016) and Zhou (J. Algebra \& Appl., 2022). Specifically, we prove that this class is closed under full matrix rings of any size, as well as we completely characterize when group rings over locally finite groups are generalized $\sqrt{J}$-fine. We also show that every such ring is 2-clean, thus properly placing it between generalized fine rings and 2-clean rings. Several examples are also provided to illustrate the complicated behavior of the introduced concept and its numerous boundaries.

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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).

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arXiv2026

Rings whose Non-Units are a Unit Multiple of an Element from $\sqrt{Δ(R)}$

This paper introduces and studies a new class of rings called {\it $U\sqrtΔ$-rings}. A ring $R$ is $U\sqrtΔ$ if every non-unit element can be written as the product of a unit and an element from $\sqrt{Δ(R)}$, where $\sqrt{Δ(R)}$ consists of elements some power of which lies in the special subring $Δ(R)$. We establish certain basic properties of these rings and, concretely, prove that they are simultaneously indecomposable and Dedekind-finite. We also show that the polynomial ring $R[x]$ and the Laurent polynomial ring $R[x, x^{-1}]$ are never $U\sqrtΔ$-rings, while the power series ring $R[[x]]$ inherits this property from $R$. Likewise, for left (right) Artinian rings, the conditions of being a $U\sqrtΔ$-ring and a $UN$-ring are equivalent, as well as these two conditions are preserved for the full matrix ring $M_n(R)$ of size $n\geq 1$ over $R$. In addition, for a commutative ring $R$, $M_n(R)$ is a $U\sqrtΔ$-ring exactly when $R$ is local. Furthermore, we characterize when a group ring $RG$ is a $U\sqrtΔ$-ring showing that, for a locally solvable group $G$, this occurs precisely when $R$ is a $U\sqrtΔ$-ring and $G$ is a locally finite $p$-group for some prime $p \in J(R)$.

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arXiv2026

Weakly $\sqrt{J}U$ Rings

We introduce and study the so-called {\it weakly $\sqrt{J}U$ rings} (hereafter abbreviated as {\it $W\sqrt{J}U$ rings} for short), in which every unit is of the form $j+1$ or $j-1$ for some $j$ in $\sqrt{J(R)} : = \{x \in R : x^n \in J(R) \text{ for some } n\ge 1\}$. This class of rings non-trivially generalizes the classes of $\sqrt{J}U$, $UU$, $JU$, $WUU$ and $WJU$ rings, respectively. We investigate their basic properties showing that they are Dedekind-finite, that $M_n(R)$ is never $W\sqrt{J}U$ for $n\ge 2$, and that when $\operatorname{char}(R)>0$ it must be equal to $2^α3^β$ for some $α, β\in \mathbb{N} \cup \left\{ 0 \right\}$. Moreover, for group rings $RG$, we prove that if $RG$ is $W\sqrt{J}U$, then $R$ is $W\sqrt{J}U$ and $G$ is a torsion group. In addition, when $R$ has positive characteristic and $G$ is a locally finite $p$-group, we give a complete characterization like this: $RG$ is a $W\sqrt{J}U$ ring if, and only if, either $R$ is a $\sqrt{J}U$ ring and $G$ is a $2$-group, or $R$ is a $W\sqrt{J}U$ ring with $3\in J(R)$ and $G$ is a $3$-group, or $R\cong R_1\times R_2$ with $R_1$ a $\sqrt{J}U$ ring, $R_2$ a $W\sqrt{J}U$ ring and $G$ a trivial group. Our results substantially improve on recent achievements due to Saini and Udar in Czech. Math. J. (2025).

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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

Rings Whose Units Have Identity Plus Quasi-Nilpotent Square

In this paper, we investigate the structural and characterizing properties of the so-called {\it 2-UQ rings}, that are rings such that the square of every unit is the sum of an idempotent and a quasi-nilpotent element that commute with each other. We establish some fundamental connections between 2-UQ rings and relevant widely classes of rings including 2-UJ, 2-UU and tripotent rings. Our novel results include: (1) complete characterizations of 2-UQ group rings, showing that they force underlying groups to be either 2-groups or 3-groups when $3 \in J(R)$; (2) Morita context extensions preserving the 2-UQ property when trace ideals are nilpotent; and (3) the discovery that potent 2-UQ rings are precisely the semi-tripotent rings. Furthermore, we determine how the 2-UQ property interacts with the regularity, cleanness and potent conditions. Likewise, certain examples and counter-examples illuminate the boundaries between 2-UQ rings and their special relatives. These achievements of ours somewhat substantially expand those obtained by Cui-Yin in Commun. Algebra (2020) and by Danchev {\it et al.} in J. Algebra \& Appl. (2025).

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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

On Rings with the 2-UNJ Property

In this paper, we introduce a new class of rings calling them {\it 2-UNJ rings}, which generalize the well-known 2-UJ, 2-UU and UNJ rings. Specifically, a ring $R$ is called 2-UNJ if, for every unit $u$ of $R$, the inclusion $u^2 \in 1 + Nil(R) + J(R)$ holds, where $Nil(R)$ is the set of nilpotent elements and $J(R)$ is the Jacobson radical of $R$. We show that every 2-UJ, 2-UU or UNJ ring is 2-UNJ, but the converse does {\it not} necessarily hold, and we also provide counter-examples to demonstrate this explicitly. We, moreover, investigate the connections between these rings and other algebraic properties such as being potent, tripotent, regular and exchange rings, respectively. In particular, we thoroughly study some natural extensions, like matrix rings and Morita contexts, obtaining new characterizations that were not addressed in previous works. Furthermore, we establish conditions under which group rings satisfy the 2-UNJ property. These results not only provide a better understanding of the structure of 2-UNJ rings, but also pave the way for future intensive research in this area.

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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

A New Characterization of Semi-Tripotent Rings

We give a comprehensive study of the so-called \textit{semi-tripotent rings} obtaining their new and non-trivial characterization as well as a complete description in terms of sums and products of some special elements. Particularly, we explore in-depth when a group ring is semi-tripotent. Our results somewhat supply those established by Ko$ş$an et al. in Can. Math. Bull. (2019).

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arXiv2025

On Strongly $Δ$-Clean Rings

This study explores in-depth the structure and properties of the so-called {\it strongly $Δ$-clean rings}, that is a novel class of rings in which each ring element decomposes into a sum of a commuting idempotent and an element from the subset $Δ(R)$. Here, $Δ(R)$ stands for the extension of the Jacobson radical and is defined as the maximal subring of $J(R)$ invariant under the unit multiplication. We present a systematic framework for these rings by detailing their foundational characteristics and algebraic behavior under standard constructions, as well as we explore their key relationships with other well-established ring classes. Our findings demonstrate that all strongly $Δ$-clean rings are inherently strongly clean and $ΔU$, but under centrality constraints they refine the category of uniquely clean rings. Additionally, we derive criteria for the strong $Δ$-clean property in triangular matrix rings, their skew analogs, trivial extensions, and group rings. The analysis reveals deep ties to boolean rings, local rings, and quasi-duo rings by offering new structural insights in their algebraic characterization.

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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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arXiv2025

Rings in which all elements are the sum of a central element and an element from $Δ(R)$

We define and consider in-depth the so-called $CΔ$ rings as those rings $R$ whose elements are a sum of an element in $C(R)$ and of an element in $Δ(R)$. Our achieved results somewhat strengthen these recently obtained by Ma-Wang-Leroy in Czechoslovak Math. J. (2024) as well as these due to Kurtulmaz-Halicioglu-Harmanci-Chen in Bull. Belg. Math. Soc. Simon Stevin (2019). Specifically, we succeeded to establish that exchange $CΔ$ rings are always clean as well as that exchange CN rings are strongly clean. Likewise, we prove that, for any ring $R$, the ring of formal power series $R[[x]]$ over $R$ is $CΔ$ if, and only if, so is $R$. And, furthermore, we show that, for any ring $R$, if the polynomial ring $R[x]$ is a $CΔ$ ring, then $R$ satisfies the Köthe conjecture. Some other closely related things concerning certain extensions of $CΔ$ rings are also presented.

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arXiv2025

Generalizing Semi-$n$-Potent Rings

We define and explore the class of rings $R$ for which each element in $R$ is a sum of a tripotent element from $R$ and an element from the subring $Δ(R)$ of $R$ which commute each other. Succeeding to obtain a complete description of these rings modulo their Jacobson radical as the direct product of a Boolean ring and a Yaqub ring, our results somewhat generalize those established by Koşan-Yildirim-Zhou in Can. Math. Bull. (2019).

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arXiv2025

Rings with 2-$Δ$U property

Rings in which the square of each unit lies in $1+Δ(R)$, are said to be $2$-$ΔU$, where $J(R)\subseteqΔ(R) =: \{r \in R | r + U(R) \subseteq U(R)\}$. The set $Δ(R)$ is the largest Jacobson radical subring of $R$ which is closed with respect to multiplication by units of $R$ and is studied in \cite{2}. The class of $2$-$ΔU$ rings consists several rings including $UJ$-rings, $2$-$UJ$ rings and $ΔU$-rings, and we observe that $ΔU$-rings are $UUC$. The structure of $2$-$ΔU$ rings is studied under various conditions. Moreover, the $2$-$ΔU$ property is studied under some algebraic constructions.

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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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arXiv2024

Rings Whose Non-Invertible Elements are Weakly Nil-Clean

In regard to our recent studies of rings with (strongly, weakly) nil-clean-like properties, we explore in-depth both the structural and characterization properties of those rings whose elements that are not units are weakly nil-clean. Group rings of this sort are considered and described as well.

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