Abstract
Let $R$ be any ring with identity. An element $a \in R$ is called nil-clean, if $a=e+n$ where $e$ is an idempotent element and $n$ is a nil-potent element. In this paper we give necessary and sufficient conditions for a $2\times 2$ matrix over an integral domain $R$ to be nil-clean.
Highlights
Let R be an associative ring with identity
An element a ∈ R is called clean if a = e + u, where e, u ∈ R, e is an idempotent and u is a unit
A is called strongly clean if it has a clean representation in which the idempotent and the unit commute
Summary
Let R be an associative ring with identity. An element a ∈ R is called clean if a = e + u, where e, u ∈ R, e is an idempotent and u is a unit. A ring R is nil-clean if each of its elements is the sum of an idempotent and a nil-potent. If ab cd is nil-clean as above, it has trace 1 and the equation (1) has a solution x, y, w with x(1 − x) = yw We shall show these two conditions characterize a nil-clean matrix [see: Theorem 3.1]. It has been shown [[7], Cor. 5.4] that central idempotents and central nil-potents in a ring are uniquely nil-clean. Let Rn be the free R-module of n tuples of elements of R
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Journal of Algebra Combinatorics Discrete Structures and Applications
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
