Peter Vassilev Danchev

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• Doctor of Mathematics
• Professor (Full) at Bulgarian Academy of Sciences

Continuing recent studies of both the hereditary and super properties of certain classes of Abelian groups, we explore in-depth what is the situation in the quite large class consisting of directly finite Abelian groups. Trying to connect some of these classes, we specifically succeeded to prove the surprising criteria that a relatively Hopfian gro...

We study the problem of when a periodic square matrix of order $n\times n$ over an arbitrary field $\mathbb{F}$ is decomposable into the sum of a square-zero matrix and a torsion matrix and show that this decomposition can always be obtained for matrices of rank at least $\frac{n}{2}$ when $\mathbb{F}$ is either a field of prime characteristic, or...

We classify those finite fields $\mathbb{F}_q$, for $q$ a power of some fixed prime number, whose members are the sum of an $n$-potent element with $n>1$ and a 4-potent element. It is shown that there are precisely ten non-trivial pairs $(q,n)$ for which this is the case. This continues a recent publication by Cohen-Danchev et al. in Turk. J. Math....

We define and consider in-depth the so-called "C∆ rings" as those rings R whose elements are a sum of an element in C(R) and of an element in ∆(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. Sim...

We study the problem of when a periodic square matrix of order n over an arbitrary field F is decomposable into the sum of a square-zero matrix and a torsion matrix, and show that this decomposition can always be obtained for matrices of rank at least n 2 when F is either a field of prime characteristic, or the field of rational numbers, or an alge...

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 ∆(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 gene...

By defining the classes of generalized co-Hopfian and relatively co-Hopfian groups, respectively, we consider two expanded versions of the generalized co-Bassian groups and of the classical co-Hopfian groups giving a close relationship with them. Concretely, we completely describe generalized co-Hopfian p-groups for some prime p obtaining that such...

Rings in which the square of each unit lies in $1+\Delta(R)$, are said to be $2$-$\Delta U$, where $J(R)\subseteq\Delta(R) =: \{r \in R | r + U(R) \subseteq U(R)\}$. The set $\Delta (R)$ 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 $2$-$\Delta...

We prove in this short note that any uniformly totally inert subgroup is commensurable with a strongly invariant subgroup, thus somewhat extending results due to Chekhlov-Danchev in Rocky Mount.

Rings in which the square of each unit lies in 1 + ∆(R), are said to be 2-∆U , where J(R) ⊆ ∆(R) =: {r ∈ R|r + U (R) ⊆ U (R)}. The set ∆(R) is the largest Jacobson radical subring of R which is closed with respect to multiplication by units of R and is studied in [19]. The class of 2-∆U rings consists several rings including U J-rings, 2-U J rings...

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 $\Delta(R)$ of $R$ for every unit $u\in R$. We succeeded to describe for any $n\geq 1$ all reduced $\pi$-regular $(2n-1)$-$\Delta$U rings by showing that they satisfy the equation $x^{2n}=x$ as well as to prove that the property of bei...

Some variations of \pi -regular and nil clean rings were recently introduced in the works of the first author: “A generalization of \pi -regular rings, Turkish J. Math. 43 (2), 702–711 (2019)”, “A symmetrization in \pi -regular rings, Trans. A. Razmadze Math. Inst. 174 (3), 271–275 (2020)”, “A symmetric generalization of \pi -regular rings, Ric. Ma...

We study in-depth those rings R for which, there exists a fixed n ≥ 1, such that u^n − 1 lies in the subring ∆(R) of R for every unit u ∈ R. We succeeded to describe for any n ≥ 1 all reduced π-regular (2n − 1)-∆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 tantamo...

Westudythoseringsinwhichallinvertible elements are weakly nil-clean, calling them 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 UNC rings. Specifically, our main achievements are that the triangular matr...

We define the class of CUSC rings, which are rings whose clean elements are uniquely strongly clean. These rings are a common generalization of the so-called 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 generalize the so-called CUC rings, define...

We consider and study those rings in which each nil-clean or clean element is uniquely nil-clean. We establish that those rings are abelian. More precisely, 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 uniquel...

As a common non-trivial generalization of the concept of a (proper) generalized Bassian group, we introduce the notion of a semi-generalized Bassian group and initiate its comprehensive investigation. Precisely, we give a satisfactory characterization of these groups by showing in the cases of 𝑝-torsion groups, torsion-free groups and splitting mix...

We continue the study in-depth of the so-called [Formula: see text]-UU rings for any [Formula: see text], that were defined by the first-named author in [Danchev, On exchange P -UU unital rings, Toyama Math. J. 39(1) (2017) 1–7] as those rings [Formula: see text] for which [Formula: see text] is always a nilpotent for every unit [Formula: see text]...

We significantly strengthen results on the structure of matrix rings over finite fields and apply them to describe the structure of the so-called weakly n-torsion clean rings. Specifically, we establish that, for any field F with either exactly seven or strictly more than nine elements, each matrix over F is presentable as a sum of of a tripotent m...

We investigate the notion of semi-nil clean rings, defined as those rings in which each element can be expressed as a sum of a periodic and a nilpotent element. Among our results, we show that if R is a semi-nil clean NI ring, then R is periodic. Additionally, we demonstrate that every group ring RG of a nilpotent group G over a weakly 2-primal rin...

This paper targets to generalize the notion of Hopfian groups in the commutative case by defining the so-called relatively Hopfian groups and weakly Hopfian groups, and establishing some their crucial properties and characterizations. Specifically, we prove that for a reduced Abelian p-group G such that p ω G is Hopfian (in particular, is finite),...

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.

We study algebras satisfying a two-term multilinear identity of the form x 1 · · · x n = qx σ(1) · · · x σ(n). We show that such algebras with q = 1 are eventually commutative in the sense that the equality x 1 · · · x k = x τ (1) · · · x τ (k) holds for k large enough and all permutations τ ∈ S k. For various natural examples, we prove that k can...

Suppose R is a commutative unitary ring of prime charac-teristic p and G is a multiplicative abelian group. The cardinality of the set id(RG) consisting of all idempotent elements in the group ring RG, is explicitly calculated only in terms associated with R and G or their sections.

In the this paper, as a generalization of the classical periodic rings, we explore those rings whose elements are additively generated by two (or more) periodic elements by calling them additively periodic. We prove that, in some major cases, additively periodic rings remain periodic too. This includes, for instance, algebraic algebras, group rings...

We systematically study those rings whose non-units are a sum of an idempotent and a nilpotent. Some crucial characteristic properties are completely described as well as some structural results for this class of rings are obtained. This work somewhat continues two publications on the subject due to Diesl (J. Algebra, 2013) and Karimi-Mansoub et al...

In connection to two recent publications of ours in Arch. Math. Basel (2021) and Acta Math. Hungar. (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...

As a common nontrivial generalization of the notion of a generalized co-Bassian group, recently defined by the third author, we introduce the notion of a semi-generalized co-Bassian group and initiate its comprehensive study. Specifically, we give a complete characterization of these groups in the cases of [Formula: see text]-torsion groups and gro...

We consider in-depth and characterize in certain aspects those rings whose non-units are strongly nil-clean in the sense that they are a sum of commuting nilpotent and idempotent. In addition, we examine those rings in which the non-units are uniquely nil-clean in the sense that they are a sum of a nilpotent and an unique idempotent. In fact, we su...

In this paper, we prove that every matrix over a division ring is representable as a product of at most 10 traceless matrices as well as a product of at most four semi-traceless matrices. By applying this result and the obtained so far other results, we show that elements of some algebras possess some rather interesting and nontrivial decomposition...

We study the problem of when a periodic square matrix over a field F is decomposable into the sum of a square-zero matrix and a torsion matrix, and show that this decomposition can always be obtained when F is either a field of prime characteristic, the field of rational numbers or an algebraically closed field of zero characteristic. We also provi...

Let F be a field. We show that given any polynomial q(x) ∈ F[x] of degree n and any matrix A consisting in its main diagonal of k 0-blocks of order one with k < n − k, and an invertible non-derogatory block of order n − k whose characteristic polynomial has the same trace as q(x), we can construct a square-zero matrix N such that the characteristic...

In connection with the work of Malev published in [J. Algebra Appl. 13 (2014) 1450004; J. Algebra Appl. 20 (2021) 2150074], we continue to provide a classification of possible images of multilinear polynomials on generalized quaternion algebras.

In this paper, we define and explore in-depth the notion of 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 w...

We define and explore in-depth the notion of 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 (of-ten named directly finite) as well as that, f...

We study those rings in which all invertible elements are weakly nil-clean calling them 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 UNC rings. Specifically , our main achievements are that the triangul...

We define the class of CUSC rings, that are those rings whose clean elements are uniquely strongly clean. These rings are a common generalization of the so-called 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 CUC ring...

We introduce the concept of ACP-Baer rings, and investigate its properties. We say a ring R is right ACP-Baer if the right annihilator of every cyclic projective right R-module in R is pure as a right ideal. This class of rings generalizes the class of right APP-rings (Z. Liu and R. Zhao, A generalization of PP-rings and p.q.-Baer rings, Glasg. Mat...

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. Debrece...

Some variations of π-regular and nil clean rings were recently introduced in [5, 7, 6], respectively. In this paper, we examine the structure and relationships between these classes of rings. Specifically, we prove that (m, n)-regularly nil clean rings are left-right symmetric and also show that the inclusions (D-regularly nil clean) ⊆ (regularly n...

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 ∈ R for which aa D ⃝ = bb D ⃝ are provided, and some new equivalent conditions on pseudo n-strong Drazin inverses are obtained. In particular, we show that an element a ∈ R is pseudo n-strong...

We continue the study in-depth of the so-called n-UU rings for any n ≥ 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 ∈ R. Specifically, for any n ≥ 2, we prove that a ring is strongly n-nil-clean if, and only if, it is simultaneously strongly π-regu...

In the present paper, as a generalization of the classical periodic rings, we explore those rings whose elements are additively generated by two (or more) periodic elements by calling them additively periodic. We prove that, in some major cases, additively periodic rings remain periodic too; this includes, for instance, algebraic algebras, group ri...

We continue the study in-depth of the so-called n-UU rings for any n ≥ 1, defined by the first-named author in Toyama Math. J. (2017). Specifically, for any n ≥ 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 Abyzov in Sib. Math. J. (...

As a common non-trivial generalization of the notion of a generalized co-Bassian group, recently defined by the third author, we introduce the notion of a semi-generalized co-Bassian group and initiate its comprehensive study. Specifically, we give a complete characterization of these groups in the cases of p-torsion groups and groups of finite tor...

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 p...

The main goal of this paper is to extend [J. Algebra Appl. 20 (2021), 2150074] to generalized quaternion algebras, even when these algebras are not necessarily division rings. More precisely, in such cases, the image of a multilinear polynomial evaluated on a quaternion algebra is a vector space and we additionally provide a classification of possi...

The main goal of this paper is to extend [J. Algebra Appl. 20 (2021), 2150074] to generalized quaternion algebras, even when these algebras are not necessarily division rings. More precisely, in such cases, the image of a multilinear polynomial evaluated on a quaternion algebra is a vector space and we additionally provide a classification of possi...

As a common non-trivial generalization of the concept of a proper generalized Bassian group, we introduce the notion of a semi-generalized Bassian group and initiate its comprehensive investigation. Precisely, we give a satisfactory characterization of these groups by showing in the cases of p-torsion groups, torsion-free groups and splitting mixed...

For any $n\ge 2$ and fixed $k\ge 1$, we give necessary and sufficient conditions for an arbitrary nonzero square matrix in the matrix ring $\mathbb{M}_n(\mathbb{F})$ to be written as a sum of an invertible matrix $U$ and a nilpotent matrix $N$ with $N^k=0$ over an arbitrary field $\mathbb{F}$.

We study the problem when every matrix over a division ring is representable as either the product of traceless matrices or the product of semi-traceless matrices, and also give some applications of such decompositions. Specifically, we establish the curious facts that every matrix over a division ring is a product of at most twelve traceless matri...

A famous conjecture states that any uniformly fully inert subgroup of a given group is commensurable with a fully invariant subgroup (see [3] and [4]). In this short note, we settle this problem completely in the affirmative for an arbitrary Abelian group.

For n ≥ 2 and fixed k ≥ 1, we study when an endomorphism f of F n , where F is an arbitrary field, can be decomposed as t + m where t is a root of the unity endomorphism and m is a nilpotent endomorphism with m k = 0. For fields of prime characteristic, we show that this decomposition holds as soon as the characteristic polynomial of f is algebraic...

We consider two variants of those Abelian groups with all proper strongly invariant subgroups isomorphic and give an in-depth study of their basic and specific properties in either parallel or contrast to the Abelian groups with all proper fully invariant (respectively, characteristic) subgroups isomorphic, which are studied in details by the curre...

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...

We define and study the so-called CI-extending property for Abelian groups, thus somewhat extending results due to Birkenmeier et al. in Commun. Algebra (2001) concerning the FI-extending property.

A ring is called UU if each its unit is a unipotent. We prove that the group ring R[G] is a commutative UU ring if, and only if, R is a commutative UU ring and G is an abelian 2-group. This extends a result due to McGovern-Raja-Sharp (J. Algebra Appl., 2015) established for commutative nil- clean group rings. In some special cases we also discover...

We define and study the so-called CI-extending property for Abelian groups, thus somewhat extending results due to Birkenmeier et al. in Commun. Algebra (2001) concerning the FI-extending property.

Let F be an algebraically closed field and let R be a locally finite algebra over F . This paper aims to show that any element of R is a product of at most three unipotent elements from R if and only if the element lies in the first derived subgroup of the unit group of R . In addition, this necessary and sufficient condition is applied to twisted...

It is known that a mixed abelian group G with torsion T is Bassian if, and only if, it has finite torsion-free rank and has finite p-torsion (i.e., each T p is finite). It is also known that if G is generalized Bassian, then each pT p is finite, so that G has bounded p-torsion. To further describe the generalized Bassian groups, we start by charact...

We significantly strengthen results on the structure of matrix rings over finite fields and apply them to describe the structure of the so-called weakly n-torsion clean rings. Specifically, we establish that, for any field F with either exactly seven or strictly more than nine elements, each matrix over F is presentable as a sum of of a tripotent m...

We significantly strengthen results on the structure of matrix rings over finite fields and apply them to describe the structure of the so-called weakly n-torsion clean rings. Specifically, we establish that, for any field F with either exactly seven or strictly more than nine elements, each matrix over F is presentable as a sum of of a tripotent m...

For n ≥ 2 and fixed k ≥ 1, we study when a square matrix A over an arbitrary field 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 ∈ Mn(F) is algebraic over its base field and the...

For any $n\ge 2$ and fixed $k\ge 1$, we give necessary and sufficient conditions for an arbitrary nonzero square matrix in the matrix ring $\mathbb{M}_n(\mathbb{F})$ to be written as a sum of an invertible matrix $U$ and a nilpotent matrix $N$ with $N^k=0$ over an arbitrary field $\mathbb{F}$.

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 mor...

We consider two variants of those Abelian groups with all proper characteristic subgroups isomorphic and give an in-depth study of their basic and specific properties in either parallel or contrast to the Abelian groups with all proper fully invariant subgroups isomorphic, which are studied in details by the current authors in Commun. Algebra (2015...

We explore the situation where all companion n × 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,...

For Abelian p-groups, Goldsmith, Salce, et al., introduced the notion of minimal full inertia. In parallel to this, we define the concept of minimal characteristic inertia and explore those p-primary Abelian groups having minimal characteristic inertia. We establish the surprising result that, for each Abelian p-group A, the square A⊕A has the mini...

In this paper we show that any matrix A in Mn(F) over an arbitrary field F can be decomposed as a sum of an invertible matrix and a nilpotent matrix of order at most two if and only if its rank is at least n 2. We also study when A can be decomposed as the sum of a torsion matrix and a nilpotent matrix of order at most two. For fields of prime char...

Let $F$ be a field with at least three elements and $G$ a locally finite group. This paper aims to show that if either $F$ is algebraically closed or the characteristic of $F$ is positive, then an element in the group algebra $FG$ is a product of unipotent elements if, and only if, it? lies in the first derived subgroup of the unit group of $FG$. I...

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 origi...

Recall that a ring R is called strongly pi-regular if, for every a in R, there is a positive integer n 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...

For Abelian p-groups, Goldsmith, Salce, et al., introduced the notion of minimal full inertia. In parallel to this, we define the concept of minimal characteristic inertia and explore those p-primary Abelian groups having minimal characteristic inertia. We establish the surprising result that, for each Abelian p-group A, the square A ⊕ A has the mi...

The property that we have termed generalized Bassian is a natural concept for many areas of algebra, namely the existence of an injective homo-morphism A → A/I for an object (group, ring, module, algebra, etc.) A with a normal sub-object I (normal subgroup, ideal, submodule, etc.) forces that I is a direct summand of A. It is a common generalizatio...

We examine those matrix rings whose entries lie in periodic rings equipped with some additional properties. Specifically, we prove that the full n × n matrix ring M n (R) is periodic for all natural numbers n, whenever the base ring R is weakly 2-primal periodic. This somewhat substantially refines recent results in the subject established by Cui-D...

In regard to our recent paper in J. Algebra (2021), we define and explore the classes of universally fully transitive and universally Krylov transitive torsion-free Abelian groups. A characterization theorem is proved in which numerous interesting properties of such groups are demonstrated. In addition, we prove the curious fact that these two clas...

We study some close relationships between the classes of transitive, fully transitive and Krylov transitive torsion-free Abelian groups. In addition, as an application of the achieved assertions, we resolve some old-standing problems, posed by Krylov, Mikhalev and Tuganbaev in their monograph [P. A. Krylov, A. V. Mikhalev and A. A. Tuganbaev, Endom...

In order to find a suitable expression of an arbitrary square matrix over an arbitrary finite commutative ring, we prove that every such 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 Linear Multili...

We calculate the probability when a finite commutative ring is weakly nil-clean in terms of invariants associated only with the given whole ring. This continues the study of our recent paper concerned with the nil-clean case [P. Danchev and M. Samiei, The probability when a finite commutative ring is nil-clean, Trans. A. Razmadze Math. Inst. 176 (2...

We examine those rings in which the elements are sums or differences of nilpotents and potents (also including in some special cases tripotents). Such decompositions of matrices over certain rings and fields are also studied. These results of ours somewhat support recent achievements presented in a publication due to Abyzov-Tapkin (Siber. Math. J.,...

We obtain a new and non-trivial characterization of periodic rings (that are those rings $R$ for which, for each element $x$ in $R$, there exists two different integers $m$, $n$ strictly greater than $1$ with the property $x^m=x^n$) in terms of nilpotent elements which supplies recent results in this subject by Cui--Danchev published in (J. Algebra...

We examine those rings in which the elements are sums or differences of nilpotents and potents (also including in some special cases tripotents). Such decompositions of matrices over certain rings and fields are also studied. These results of ours somewhat support recent achievements presented in a publication due to Abyzov-Tapkin (Siber. Math. J.,...

We study some close relationships between the classes of transitive , fully transitive and Krylov transitive torsion-free Abelian groups. In addition, as an application of the achieved assertions, we resolve some old-standing problems, posed by Krylov-Mikhalev-Tuganbaev in their monograph [17]. Specifically, we answer Problem 44 from there in the a...

We study when every square matrix over an algebraically closed field or over a finite field is decomposable into a sum of a potent matrix and a nilpotent matrix of order 2. This can be related to our recent paper, published in Linear & Multilinear Algebra (2022). We also completely address the question when each square matrix over an infinite field...

We will compute the probability when a finite commutative ring is either nil-neat or weakly nil-neat, respectively.

We calculate the probability when a finite commutative ring is weakly nil-clean in terms of invariants associated only with the given whole ring. This continues the study of our recent paper concerned with the nil-clean case.

An element v of an arbitrary ring R is called an involution if v^2 = 1 and a quasi-involution if either v or 1 − v is an involution. We thereby define R to be quasi invo-clean as the one whose elements are written in the form of a sum of an idempotent and a quasi-involution. This considerably extends the class of invo-clean rings introduced by the...

Let [Formula: see text] be a ring and let [Formula: see text] be an arbitrary but fixed positive integer. We characterize those rings [Formula: see text] whose elements [Formula: see text] satisfy at least one of the relations that [Formula: see text] or [Formula: see text] is a nilpotent whenever [Formula: see text]. This extends results from the...