Searcharxiv⌕ Search

arXiv subjects

Peter Danchev

Publications and source records attributed to Peter Danchev.

At least 55 records · Page 3Linked to original sources
arXiv2024

Rings with $u-1$ Quasinilpotent for Each Unit $u$

We define and explore in-depth the notion of {\it UQ rings} by showing their important properties and by comparing their behavior with that of the well-known classes of UU rings and JU rings, respectively. Specifically, among the other established results, we prove that UQ rings are always Dedekind finite (often named directly finite) as well as that, for semipotent rings $R$, the following equivalence hold: $R/J(R)$ is UQ $\iff$ $R$ is UQ having the property that the set $QN(R)$ of quasinilpotent elements of $R$ coincides with the Jacobson radical $J(R)$ of $R$.

math.RA↗
arXiv2024

Rings With $u^n-1$ Nilpotent For Each Unit $u$

We continue the study in-depth of the so-called $n$-UU rings for any $n\geq 1$, that were defined by the first-named author in Toyama Math. J. (2017) as those rings $R$ for which $u^n-1$ is always a nilpotent for every unit $u\in R$. Specifically, for any $n\geq 2$, we prove that a ring is strongly $n$-nil-clean if, and only if, it is simultaneously strongly $π$-regular and an $(n-1)$-UU ring. This somewhat extends results due to Diesl in J. Algebra (2013), Abyzov in Sib. Math. J. (2019) and Cui-Danchev in J. Algebra Appl. (2020). Moreover, our results somewhat improves the ones obtained by Ko$ş$an et al. in Hacettepe J. Math. Stat. (2020).

math.RA↗
arXiv2024

Rings Whose Invertible Elements Are Weakly Nil-Clean

We study those rings in which all invertible elements are weakly nil-clean calling them {\it UWNC rings}. This somewhat extends results due to Karimi-Mansoub et al. in Contemp. Math. (2018), where rings in which all invertible elements are nil-clean were considered abbreviating them as {\it UNC rings}. Specifically, our main achievements are that the triangular matrix ring ${\rm T}_n(R)$ over a ring $R$ is UWNC precisely when $R$ is UNC. Besides, the notions UWNC and UNC do coincide when $2 \in J(R)$. We also describe UWNC $2$-primal rings $R$ by proving that $R$ is a ring with $J(R) = {\rm Nil}(R)$ such that $U(R)=\pm 1+{\rm Nil}(R)$. In particular, the polynomial ring $R[x]$ over some arbitrary variable $x$ is UWNC exactly when $R$ is UWNC. Some other relevant assertions are proved in the present direction as well.

math.RA↗
arXiv2024

Rings Whose Clean Elements Are Uniquely Strongly Clean

We define the class of {\it CUSC} rings, that are those rings whose clean elements are uniquely strongly clean. These rings are a common generalization of the so-called {\it USC} rings, introduced by Chen-Wang-Zhou in J. Pure \& Applied Algebra (2009), which are rings whose elements are uniquely strongly clean. These rings also generalize the so-called {\it CUC} rings, defined by Calugareanu-Zhou in Mediterranean J. Math. (2023), which are rings whose clean elements are uniquely clean. We establish that a ring is USC if, and only if, it is simultaneously CUSC and potent. Some other interesting relationships with CUC rings are obtained as well.

math.RA↗
arXiv2024

Rings Whose Clean and Nil-Clean Elements Have Some Clean-Like Properties

We define two types of rings, namely the so-called CSNC and NCUC that are those rings whose clean elements are strongly nil-clean, respectively, whose nil-clean elements are uniquely clean. Our results obtained in this paper somewhat expand these obtained by Calugareanu-Zhou in Mediterr. J. Math. (2023) and by Cui-Danchev-Jin in Publ. Math. Debrecen (2024), respectively.

math.RA↗
arXiv2023

Some New Results on Pseudo n-Strong Drazin Inverses in Rings

In this paper, we give a further study in-depth of the pseudo $n$-strong Drazin inverses in an associative unital ring $R$. The characterizations of elements $a,b\in R$ for which $aa^{\tiny{\textcircled{\qihao D}}}=bb^{\tiny{\textcircled{\qihao D}}}$ are provided, and some new equivalent conditions on pseudo $n$-strong Drazin inverses are obtained. In particular, we show that an element $a\in R$ is pseudo $n$-strong Drazin invertible if, and only if, $a$ is $p$-Drazin invertible and $a-a^{n+1}\in \sqrt{J(R)}$ if, and only if, there exists $e^2=e\in {\rm comm}^2(a)$ such that $ae\in \sqrt{J(R)}$ and $1-(a+e)^n\in \sqrt{J(R)}$. We also consider pseudo $n$-strong Drazin inverses with involution, and discuss the extended versions of Cline's formula and Jacobson's lemma of this new class of generalized inverses. Likewise, we define and explore the so-called {\it pseudo $π$-polar} rings and demonstrate their relationships with periodic rings and strongly $π$-regular rings, respectively.

math.RA↗
arXiv2023

Supper Bassian and Nearly Generalized Bassian Abelian Groups

In connection to two recent publications of ours in Arch. Math. Basel (2021) and Acta Math. Hung. (2022), respectively, and in regard to the results obtained in Arch. Math. Basel (2012), we have the motivation to study the near property of both Bassian and generalized Bassian groups. Concretely, we prove that if an arbitrary reduced group has the property that all of its proper subgroups are generalized Bassian, then it is also generalized Bassian itself, that is, if G is an arbitrary nearly generalized Bassian group, then either G is a quasi-cyclic group or G is generalized Bassian of finite torsion-free rank. Moreover, we show that there is a complete description of the so-called super Bassian groups, that are those groups whose epimorphic images are all Basian, and give their full characterization. Also, we establish that the hereditary property of Bassian groups gives nothing new as it coincides with the ordinary Bassian property.

math.GR↗
arXiv2023

Rings whose Nil-Clean and Clean Elements are Uniquely Nil-Clean

We consider and study those rings in which each nil-clean or clean element is uniquely nil-clean. We establish that, for abelian rings, these rings have a satisfactory description and even it is shown that the classes of abelian rings and the rings in which nil-clean elements are uniquely nil-clean do coincide. Moreover, we prove that the rings in which clean elements are uniquely nil-clean coincide with the subclass of abelian rings consisting of only unipotent units and, in particular, that in the semipotent case we have a complete characterization only in terms of the former ring and its divisions. Likewise, some extension properties and group rings for such kinds of rings are also considered.

math.RA↗
arXiv2023

Decompositions of Matrices Into a Sum of Torsion Matrices and Matrices of Fixed Nilpotence

For $n\ge 2$ and fixed $k\ge 1$, we study when a square matrix $A$ over an arbitrary field $\mathbb{F}$ can be decomposed as $T+N$ where $T$ is a torsion matrix and $N$ is a nilpotent matrix with $N^k=0$. For fields of prime characteristic, we show that this decomposition holds as soon as the characteristic polynomial of $A\in \mathbb{M}_{n}(\mathbb{F})$ is algebraic over its base field and the rank of $A$ is at least $\frac nk$, and we present several examples that show that the decomposition does not hold in general. Furthermore, we completely solve this decomposition problem for $k=2$ and nilpotent matrices over arbitrary fields (even over division rings). This somewhat continues our recent publications in Lin. \& Multilin. Algebra (2023) and Internat. J. Algebra \& Computat. (2022) as well as it strengthens results due to Calugareanu-Lam in J. Algebra \& Appl. (2016).

math.RA↗
arXiv2023

On Tight Submodules of Modules over Valuation Domains

This note offers an unusual approach of studying a class of modules inasmuch as it is investigating a subclass of the category of modules over a valuation domain. This class is far from being a full subcategory, it is not even a category. Our concern is the subclass consisting of modules of projective dimension not exceeding one, admitting only morphisms whose kernels and cokernels are also objects in this subclass. This class is still tractable, several features are in general simpler than in module categories, but lots of familiar properties are lost. A number of results on modules in this class are similar to those on modules over rank one discrete valuation domains (where the global dimension is 1). The study is considerably simplified by taking advantage of the general theory of modules over valuation domains available in the literature, e.g. in [14]-[15]. Our main goal is to establish the basic features and have a closer look at injectivity, pure-injectivity, and cotorsionness, but we do not wish to enter into an in-depth study of these properties.

math.AC↗
arXiv2023

Companion Weakly Periodic Matrices over Finite and Countable Fields

We explore the situation where all companion $n \times n$ matrices over a field $F$ are weakly periodic of index of nilpotence $2$ and prove that this can be happen uniquely when $F$ is a countable field of positive characteristic, which is an algebraic extension of its minimal simple (finite) subfield, with all subfields of order greater than $n$. In particular, in the commuting case, we show even that $F$ is a finite field of order greater than $n$. Our obtained results somewhat generalize those obtained by Breaz-Modoi in Lin. Algebra & Appl. (2016).

math.RA↗
arXiv2022

On Strongly pi-Regular Rings with Involution

Recall that a ring R is called strongly pi-regular if, for every a in R, there is a positive integer n, depending on a, such that a^n belongs to the intersection of a^{n+1}R and Ra^{n+1}. In this paper we give a further study of the notion of a strongly pi-star-regular ring, which is the star-version of strongly pi-regular rings and which was originally introduced by Cui-Wang in J. Korean Math. Soc. (2015). We also establish various properties of these rings and give several new characterizations in terms of (strong) pi-regularity and involution. Our results also considerably extend recent ones in the subject due to Cui-Yin in Algebra Colloq. (2018) proved for pi-star-regular rings and due to Cui-Danchev in J. Algebra Appl. (2020) proved for star-periodic rings.

math.RA↗
arXiv2021

Decompositions of Matrices into Potent and Square-Zero Matrices

In order to find a suitable expression of an arbitrary square matrix over an arbitrary finite commutative ring, we prove that every such a matrix is always representable as a sum of a potent matrix and a nilpotent matrix of order at most two when the Jacobson radical of the ring has zero-square. This somewhat extends results of ours in Lin. & Multilin. Algebra (2021) established for matrices considered on arbitrary fields. Our main theorem also improves on recent results due to Abyzov et al. in Mat. Zametki (2017), Šter in Lin. Algebra & Appl. (2018) and Shitov in Indag. Math. (2019).

math.RA↗
arXiv2019

Commutative Weakly Nil-Neat Group Rings

Let R be a ring and let G be a group. We prove a rather curious necessary and sufficient condition for the commutative group ring RG to be weakly nil-neat only in terms of R,G and their sections. This somewhat expands three recent results, namely those established by McGovern et al. in (J. Algebra Appl., 2015), by Danchev-McGovern in (J. Algebra, 2015) and by the present authors in (J. Taibah Univ. Sci., 2020), related to commutative nil-clean, weakly nil-clean and nil-neat group rings, respectively.

math.AC↗
arXiv2019

n-torsion clean and almost n-torsion clean matrix rings

We completely determine those natural numbers $n$ for which the full matrix ring $M_n(F_2)$ and the triangular matrix ring $T_n(F_2)$ over the two elements field $F_2$ are either n-torsion clean or are almost n-torsion clean, respectively. These results somewhat address and settle a question, recently posed by Danchev-Matczuk in Contemp. Math. (2019) as well as they supply in a more precise aspect the nil-cleanness property of the full matrix $n\times n$ ring $M_n(F_2)$ for all naturals $n \geq 1$, established in Linear Algebra and Appl. (2013) by Breaz-Calugareanu-Danchev-Micu and again in Linear Algebra & Appl. (2018) by Ster as well as in Indag. Math. (2020) by Shitov.

math.RA↗
arXiv2018

n-Torsion Clean Rings

Let n be an arbitrary natural number. The class of (strongly) n-torsion clean rings is introduced and investigated. Abelian n-torsion clean rings are somewhat characterized and a complete characterization of strongly n-torsion clean rings is given in the case when n is odd. Some open questions are posed at the end.

math.RA↗
arXiv2015

Generalizing pi-regular rings

We introduce the class weakly nil clean rings, as rings R in which for every a\in R there exist an idempotent e and a nilpotent q such that a-e-q\in eRa. Every weakly nil clean ring is exchange. Weakly nil clean rings contain pi-regular rings as a proper subclass, and these two classes coincide in the case of central idempotents. Every weakly nil clean ring of bounded index and every weakly nil clean PI-ring is strongly pi-regular. The center of a weakly nil clean ring is strongly pi-regular, and consequently, every weakly nil clean ring is a corner of a clean ring. These results extend Azumaya [3], McCoy [24], and the second author [33] to a wider class of rings and provide partial answers to some open questions in [13] and [33]. Some other properties are also studied and several examples are given.

math.RA↗