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Peter Danchev

Publications and source records attributed to Peter Danchev.

At least 19 recordsLinked to original sources
arXiv2026

$\sqrt{\Delta}$-Fine Rings

We introduce and study the so-termed {\it $\sqrt{\Delta}$-fine rings}, a new class of rings that generalizes the classical {\it fine rings} introduced by C\u{a}lug\u{a}reanu-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 $a \in \sqrt{\Delta(R)}$. We establish that every such ring is simple, every abelian $\sqrt{\Delta}$-fine ring is indecomposable, and most notably, the matrix ring $M_n(R)$ over a $\sqrt{\Delta}$-fine ring $R$ is again $\sqrt{\Delta}$-fine for every $n \ge 1$. As a consequence, we characterize all semi-local $\sqrt{\Delta}$-fine rings as those rings which are precisely the simple Artinian rings. We also examine group rings, providing conditions under which they are either $\sqrt{\Delta}$-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 $\sqrt{\Delta}$-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 C\u{a}lug\u{a}reanu-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 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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arXiv2026

A Generalization of $\Delta$U Rings

In this paper, we introduce and study a new class of rings calling them {\it weakly $\Delta U$-rings}, hereafter abbreviated as {\it $W\Delta U$-rings} for short. A ring $R$ is said to be $W\Delta U$ if every unit of $R$ can be expressed as $\pm 1 + d$ for some $d \in \Delta(R)$, where $\Delta(R)$ is the largest Jacobson radical of $R$ that is closed under multiplication by units. Utilizing the known structure of $\Delta(R)$, we investigate the relationships between $W\Delta U$ rings and certain classical concepts such as $\Delta U$-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 $W\Delta U$ for any $n \ge 2$. We also provide complete characterizations of local, semi-local, semi-simple and semi-regular rings that are $W\Delta U$. Furthermore, it is shown for exchange rings that the $W\Delta U$ property is equivalent to being $WUJ$. Furthermore, the behavior of $W\Delta U$-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 $W\Delta U$-rings properly contains the class of $\Delta U$-rings. Finally, necessary and sufficient conditions for a group ring $RG$ to be $W\Delta U$ are established too. Resuming all of the presented above, our results expanded those by Karaba\c{c}ak et al. published in J. Algebra \& Appl. (2021).

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arXiv2026

Matrices over Finite Fields of Characteristic 2 as Sums of Diagonalizable and Square-Zero Matrices

We investigate the problem asking when any square matrix whose entries lie in a finite field of characteristic 2 is decomposable into the sum of a diagonalizable matrix and a nilpotent matrix with index of nilpotency at most 2 and, as a result, we completely resolve this question in the affirmative for any finite field of characteristic 2 having strictly more than three elements. Our main theorem of that type, combined with results from our recent publication in Linear Algebra & Appl. (2026) (see [7]), totally settle this problem for all finite fields different from $\mathbb{F}_2$ and $\mathbb{F}_3$. However, in this paper we also prove that each matrix over $\mathbb{F}_2$ is expressible as the sum of a potent matrix with index of potency not exceeding 4 and a nilpotent matrix with index of nilpotency not exceeding 2, thus substantiating recent examples due to \v{S}ter in Linear Algebra & Appl. (2018) and Shitov in Indag. Math. (2019) (see, respectively, [9] and [8]).

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arXiv2026

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

This paper introduces and studies a new class of rings called {\it $U\sqrt{\Delta}$-rings}. A ring $R$ is $U\sqrt{\Delta}$ if every non-unit element can be written as the product of a unit and an element from $\sqrt{\Delta(R)}$, where $\sqrt{\Delta(R)}$ consists of elements some power of which lies in the special subring $\Delta(R)$. 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 $U\sqrt{\Delta}$-rings, while the power series ring $R[[x]]$ inherits this property from $R$. Likewise, for left (right) Artinian rings, the conditions of being a $U\sqrt{\Delta}$-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 $U\sqrt{\Delta}$-ring exactly when $R$ is local. Furthermore, we characterize when a group ring $RG$ is a $U\sqrt{\Delta}$-ring showing that, for a locally solvable group $G$, this occurs precisely when $R$ is a $U\sqrt{\Delta}$-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 $2^\alpha 3^\beta$ for some $\alpha, \beta \in \mathbb{N} \cup \left\{ 0 \right\}$. 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

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

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 $\pi$-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 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.

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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

Some Results on Triangular Coefficient Matrix Rings

In this paper, we introduce the concept of a {\it triangular coefficient matrix ring} and investigate the structure of its ideals. We then characterize the radicals of the ring \( R_{h}[x]/\langle x^{n} \rangle \) for every positive integer \( n \), where \( R_{h}[x] \) denotes the Hurwitz polynomial ring and \( \langle x^{n} \rangle \) represents the ideal of this ring generated by \( x^{n} \). Furthermore, we explore several properties that are transferred between the base ring \( R \) and the matrix ring \( H_{n}(R) \) which is a proper subring of the triangular coefficient matrix ring.

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arXiv2025

Matrices over finite fields of odd characteristic as sums of diagonalizable and square-zero matrices

Let $\mathbb{F}$ be a finite field of odd characteristic. When $|\mathbb{F}|\ge 5$, we prove that every matrix $A$ admits a decomposition into $D+M$ where $D$ is diagonalizable and $M^2=0$. For $\mathbb{F}=\mathbb{F}_3$, we show that such decomposition is possible for non-derogatory matrices of order at least 5, and more generally, for matrices whose first invariant factor is not a non-zero trace irreducible polynomial of degree 3; we also establish that matrices consisting of direct sums of companion matrices, all of them associated to the same irreducible polynomial of non-zero trace and degree 3 over $\mathbb{F}_3$, never admit such decomposition. These results completely settle the question posed by Breaz in Lin. Algebra & Appl. (2018) asking if it is true that for big enough positive integers $n\ge 3$ all matrices $A$ over a field of odd cardinality $q$ admit decompositions of the form $E+M$ with $E^q=D$ and $M^2=0$: the answer is {\it yes} for $q\ge 5$, but there are counterexamples for $q=3$ and each order $n=3k$, $k\ge 1$.

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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$\c{s}$an et al. in Can. Math. Bull. (2019).

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