When is every matrix over a division ring a sum of an idempotent and a nilpotent?


Kosan M. T. , Lee T., Zhou Y.

LINEAR ALGEBRA AND ITS APPLICATIONS, vol.450, pp.7-12, 2014 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 450
  • Publication Date: 2014
  • Doi Number: 10.1016/j.laa.2014.02.047
  • Title of Journal : LINEAR ALGEBRA AND ITS APPLICATIONS
  • Page Numbers: pp.7-12
  • Keywords: Idempotent matrix, Nilpotent matrix, Nil-clean matrix, Matrix ring, Semilocal ring, Division ring, Strongly regular ring

Abstract

A ring is called nil-clean if each of its elements is a sum of an idempotent and a nilpotent. In response to a question of S. Breaz et al. in [1], we prove that the n x n matrix ring over a division ring D is a nil-clean ring if and only if D congruent to F-2. As consequences, it is shown that the n x n matrix ring over a strongly regular ring R is a nil-clean ring if and only if R is a Boolean ring, and that a semilocal ring R is nil-clean if and only if its Jacobson radical J (R) is nil and R/J (R) is a direct product of matrix rings over F-2. (C) 2014 Elsevier Inc. All rights reserved.