Nil-clean Rings Research Articles

Симметризация в чистых и ниль-чистых кольцах

Симметризация в чистых и ниль-чистых кольцах

Commutative feebly nil-clean group rings

Abstract An arbitrary unital ring R is called feebly nil-clean if any its element is of the form q + e − f, where q is a nilpotent and e, f are idempotents with ef = fe. For any commutative ring R and any abelian group G, we find a necessary and sufficient condition when the group ring R(G) is feebly nil-clean only in terms of R, G and their sections. Our result refines establishments due to McGovern et al. in J. Algebra Appl. (2015) on nil-clean rings and Danchev-McGovern in J. Algebra (2015) on weakly nil-clean rings, respectively.

Correction to: Weak nil clean ideals

The authors used the notion of Nil clean rings, which was introduced by P. V. Danchev and W. Wm. McGovern in their paper “Danchev P.V., McGovern W.Wm., Commutative weakly nil clean unital rings, J. Algebra 425 (2015), 410–422.”. But unfortunately, it was not cited in the paper. So the reference no. 2 of the paper should be replaced by the paper of P. V. Danchev and W. Wm. McGovern. The authors regret for the mistake and would like to thank Prof. Danchev for pointing it out.

Nil-clean elements which are not clean in certain subrings of M3(ℤ)

Let R be a ring with identity. An element x ∈ R is said to be clean if x = u + e for some unit u and idempotent e in R. If x = u + e or x = u − e for some unit u and idempotent e in R, then x is said to be weakly clean. The element x is said to be nil-clean if it is the sum of an idempotent and a nilpotent element. The ring R is called clean (weakly clean, nil-clean) if all of its elements are clean (weakly clean, nil-clean, respectively). It is known that nil-clean rings are clean. However, a nil-clean element is not necessarily clean. We give more examples of this by showing the existence of elements which are nil-clean but not clean in a subring of M3(ℤ), the ring of 3 × 3 matrices over ℤ. As a consequence, we also obtain examples of weakly clean elements which are not clean.

On Unit Nil-Clean Rings

A ring R is unit nil-clean if, for any $$a\in R$$ , there exists a unit $$u\in R$$ , such that ua is the sum of an idempotent and a nilpotent. In this paper, we completely describe the structure of unit nil-clean rings. We thereby provide a large class of rings over which every square matrix is equivalent to the sum of idempotent and nilpotent matrices. Furthermore, the uniqueness is determined by the abelian property.

Weak nil clean ideals

As a generalization of a nil clean ideal, we define a weak nil clean ideal of a ring. An ideal I of a ring R is said to be weak nil clean ideal if for any $$x\in I$$, either $$x=e+n$$ or $$x=-e+n$$, where n is a nilpotent element and e is an idempotent element of R. Some interesting properties of weak nil clean ideal and its relation with weak nil clean ring have been discussed.

Rings whose Elements are the Sum of a Tripotent and an Element from the Jacobson Radical

AbstractThis paper is about rings $R$ for which every element is a sum of a tripotent and an element from the Jacobson radical $J(R)$. These rings are called semi-tripotent rings. Examples include Boolean rings, strongly nil-clean rings, strongly 2-nil-clean rings, and semi-boolean rings. Here, many characterizations of semi-tripotent rings are obtained. Necessary and sufficient conditions for a Morita context (respectively, for a group ring of an abelian group or a locally finite nilpotent group) to be semi-tripotent are proved.

Rings additively generated by tripotents and nilpotents

A ring [Formula: see text] is Zhou nil-clean if every element in [Formula: see text] is the sum of two tripotents and a nilpotent that commute. A ring [Formula: see text] is feebly clean if for any [Formula: see text] there exist two orthogonal idempotents [Formula: see text] and a unit [Formula: see text] such that [Formula: see text]. In this paper, Zhou nil-clean rings are further discussed with an emphasis on their relations with feebly clean rings. We prove that a ring [Formula: see text] is Zhou nil-clean if and only if [Formula: see text] is feebly clean, [Formula: see text] is nil and [Formula: see text] has exponent [Formula: see text] if and only if [Formula: see text] is weakly exchange, [Formula: see text] is nil and [Formula: see text] has exponent [Formula: see text]. New properties of Zhou rings are thereby obtained.

Commutative Weakly Invo–Clean Group Rings

A ring \(R\) is called weakly invo-clean if any its element is the sum or the difference of an involution and an idempotent. For each commutative unital ring \(R\) and each abelian group \(G\), we find only in terms of \(R\), \(G\) and their sections a necessary and sufficient condition when the group ring \(R[G]\) is weakly invo-clean. Our established result parallels to that due to Danchev-McGovern published in J. Algebra (2015) and proved for weakly nil-clean rings.

A Generalization of Strongly Nil Clean Rings

Generalizing the notion of strongly nil clean rings, we introduce strongly quasi-nil clean rings. Some fundamental properties and equivalent characterizations of this class of rings are provided. By means of g-Drazin inverses, Cline’s formula and Jacobson’s lemma for strongly quasi-nil clean elements are investigated.

Structure of Zhou Nil-clean Rings

A ring R is Zhou nil-clean if every element in R is the sum of two tripotents and a nilpotent that commute. Homomorphic images of Zhou nil-clean rings are explored. We prove that a ring R is Zhou nil-clean if and only if 30 ϵ R is nilpotent and R/30R is Zhou nil-clean, if and only if R/BM(R) is 5-potent and BM(R) is nil, if and only if J(R) is nil and R/J(R) is isomorphic to a Boolean ring, a Yaqub ring, a Bell ring or a direct product of such rings. By means of homomorphic images, we completely determine when the generalized matrix ring is Zhou nil-clean. We prove that the generalized matrix ring Mn(R; s) is Zhou nil-clean if and only if R is Zhou nil-clean and s ϵ J(R).

Nil-clean rings of nilpotency index at most two with application to involution-clean rings

Nil-clean rings of nilpotency index at most two with application to involution-clean rings

English

Let R be an associative unital ring and let a ∈ R be a strongly nil clean element. We introduce a new idea for examining the properties of these elements. This approach allows us to generalize some results on nil clean and strongly nil clean rings. Also, using this technique many previous proofs can be significantly shortened. Some shorter proofs concerning nil clean elements in rings in general, and in matrix rings in particular, are presented, together with some generalizations of these results.

On expressing matrices over [formula omitted] as the sum of an idempotent and a nilpotent

We prove that for every n×n matrix A over the field Z2 there exists an idempotent matrix E such that (A−E)4=0. This result, strengthening the result of Breaz, Călugăreanu, Danchev and Micu, allows us to construct a nil-clean ring that is not strongly π-regular, which answers an open question of Diesl.

On graded nil clean rings

ABSTRACTIn this paper we introduce and study the notion of a graded (strongly) nil clean ring which is group graded. We also deal with extensions of graded (strongly) nil clean rings to graded matrix rings and to graded group rings. The question of when nil cleanness of the component, which corresponds to the neutral element of a group, implies graded nil cleanness of the whole graded ring is examined. Similar question is discussed in the case of groupoid graded rings as well.

Generalizing nil clean rings

We introduce the class of {\it unipotently nil clean} rings as these rings $R$ in which for every $a\in R$ there exist an idempotent $e$ and a nilpotent $q$ such that $a-e-1-q\in (1-e)Ra$. Each unipotently nil clean ring is weakly nil clean as well as each nil clean ring is unipotently nil clean. Our results obtained here considerably extend those from [8] and [7], respectively.

A generalization of nil-clean rings

A generalization of nil-clean rings

Strongly weakly nil-clean rings

A ring [Formula: see text] is strongly weakly nil-clean if every element in [Formula: see text] is the sum or difference of a nilpotent and an idempotent that commutes. We prove, in this paper, that a ring [Formula: see text] is strongly weakly nil-clean if and only if for any [Formula: see text], there exists an idempotent [Formula: see text] such that [Formula: see text] is nilpotent if and only if [Formula: see text] forms an ideal and [Formula: see text] is weakly Boolean if and only if [Formula: see text] or [Formula: see text], where [Formula: see text] is strongly nil-clean and [Formula: see text] is an IU ring if and only if [Formula: see text] has no homomorphic image [Formula: see text] and for any [Formula: see text], there exists [Formula: see text] (depending on [Formula: see text]) such that [Formula: see text]. These also extend known theorems of Danchev, Kosan, Zhou, etc.

Nil-clean quadratic elements

We provide a strong condition holding for nil-clean quadratic elements in any ring. In particular, our result implies that every nil-clean involution in a ring is unipotent. As a consequence, we give a complete classification of weakly nil-clean rings introduced recently in [Breaz, Danchev and Zhou, Rings in which every element is either a sum or a difference of a nilpotent and an idempotent, J. Algebra Appl. 15 (2016) 1650148, doi: 10.1142/S0219498816501486].

Matrices over Zhou nil-clean rings

ABSTRACTA ring R is Zhou nil-clean if every element in R is the sum of two tripotents and a nilpotent that commute. Let R be a Zhou nil-clean ring. If R is 2-primal (of bounded index), we prove that every square matrix over R is the sum of two tripotents and a nilpotent. These provide a large class of rings over which every square matrix has such decompositions by tripotent and nilpotent matrices.