Strongly 2-nil-clean rings

Abstract

A ring [Formula: see text] is strongly 2-nil-clean if every element in [Formula: see text] is the sum of two idempotents and a nilpotent that commute. Fundamental properties of such rings are obtained. We prove that a ring [Formula: see text] is strongly 2-nil-clean if and only if for all [Formula: see text], [Formula: see text] is nilpotent, if and only if for all [Formula: see text], [Formula: see text] is strongly nil-clean, if and only if every element in [Formula: see text] is the sum of a tripotent and a nilpotent that commute. Furthermore, we prove that a ring [Formula: see text] is strongly 2-nil-clean if and only if [Formula: see text] is tripotent and [Formula: see text] is nil, if and only if [Formula: see text] or [Formula: see text], where [Formula: see text] is a Boolean ring and [Formula: see text] is nil; [Formula: see text] is a Yaqub ring and [Formula: see text] is nil. Strongly 2-nil-clean group algebras are investigated as well.

m-nil-clean Companion Matrices

“…This is why in the proof of Theorem 3 of [2], one restricts without loss of generality to the case of companion matrices. The same technique is used in [6] to prove that M n (F 3 ) is 2-nil-clean. We are also determined to consider m-nil-clean companion matrices, based on the previously mentioned facts and on the fact that, if all companion matrices which appear in the Frobenius normal form of a matrix A are m-nil-clean, then A is also m-nil-clean.…”

“…A particular class of clean rings was introduced by Diesl in [8]: the class of rings such that all elements are sums of a nilpotent and an idempotent. Other generalizations were considered in [6] and [4]. In the former, Chen and Sheibani considered 2-nil-clean rings, rings such that all elements are 2-nil-clean, i.e., elements that are sums of two idempotents and a nilpotent element.…”

(WEAKLY) n

“…In [2], Breaz, Danchev and Zhou de…ned a ring R to be weakly nil clean if each element r 2 R can be written as r = b + e or r = b e for b 2 N (R) and e 2 Id(R). We refer the reader to [8,1,3,5,7,4,2] for a survey on nil clean and weakly nil clean rings.…”

A generalization of nil-clean rings

“…A class of strongly 2-nil-clean rings was introduced in [4]. Namely, an element a in a ring R is called strongly 2-nil-clean if it can be represented in the form a D e C f C n, where e and f are idempotents, n is a nilpotent element and they all commute with each other.…”