On the subring generated by commutators

Plnar Eroǧlu M. P.

JOURNAL OF ALGEBRA AND ITS APPLICATIONS, vol.21, no.03, 2022 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 21 Issue: 03
  • Publication Date: 2022
  • Doi Number: 10.1142/s0219498822500591
  • Journal Name: JOURNAL OF ALGEBRA AND ITS APPLICATIONS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Communication Abstracts, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
  • Keywords: Lie product, (additive) commutator, commutator ring, noncommutative ring, noncommutative polynomial
  • Dokuz Eylül University Affiliated: Yes

Abstract

Let R be a ring. By the notation [R,R] we denote the additive subgroup of R generated by all [a,b] = ab - ba in R. In this work, we partially generalize a result due to Herstein [I. N. Herstein, Topics in Ring Theory (University of Chicago Press, 1969)] showing that if 1 is an element of [R,R], then the subring generated by [R,R] is equal to R. This result implies that [R,R] cannot be a proper subring of R.