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, cilt.450, ss.7-12, 2014 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 450
  • Basım Tarihi: 2014
  • Doi Numarası: 10.1016/j.laa.2014.02.047
  • Dergi Adı: LINEAR ALGEBRA AND ITS APPLICATIONS
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Sayfa Sayıları: ss.7-12
  • Anahtar Kelimeler: Idempotent matrix, Nilpotent matrix, Nil-clean matrix, Matrix ring, Semilocal ring, Division ring, Strongly regular ring
  • Gazi Üniversitesi Adresli: Hayır

Özet

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.