Remarks on rings with u − 1 quasi-nilpotent for each unit u

Tien N. Q., KOŞAN M. T., Quynh T. C.

Journal of Algebra and its Applications, 2024 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Publication Date: 2024
  • Doi Number: 10.1142/s0219498825503177
  • 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: (semi)potent ring, nilpotent element, quasi-nilpotent element, UJ-ring, Unit element, UU-ring, ΔU-ring
  • Gazi University Affiliated: Yes

Abstract

This paper is concerned with the investigation of those rings R whose invertible elements are sums 1 + a for an element a ∈ R such that 1 − au is invertible for all invertible u ∈ R with au = ua. UJ-rings and UQ-rings (and hence UU-rings) are examples of such rings. We prove that such rings are Dedekind finite, and for semisimple rings R, R ∼= F2 × F2 × · · · × F2. In case R is also a semipotent ring, it is shown that R/J(R) is a Boolean ring.