Reversible ring property via idempotent elements


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KÖSE H., ÜNGÖR B., Harmanci A.

GEORGIAN MATHEMATICAL JOURNAL, 2022 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Publication Date: 2022
  • Doi Number: 10.1515/gmj-2022-2189
  • Journal Name: GEORGIAN MATHEMATICAL JOURNAL
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, zbMATH
  • Keywords: Reversible ring, e-reduced ring, e-symmetric ring, e-semicommutative ring, idempotent element, MODULES
  • Hacettepe University Affiliated: Yes

Abstract

Regarding the question of how idempotent elements affect the reversible property of rings, we study a version of reversibility depending on idempotents. In this perspective, we introduce right (resp., left) e-reversible rings. We show that this concept is not left-right symmetric. Basic properties of right e-reversibility in a ring are provided. Among others, it is proved that if R is a semiprime ring, then R is right e-reversible if and only if it is right e-reduced if and only if it is e-symmetric if and only if it is right e-semicommutative. Also, for a right e-reversible ring R, R is a prime ring if and only if it is a domain. It is shown that the class of right e-reversible rings lies strictly between that of e-symmetric rings and right e-semicommutative rings. Some extensions of rings are studied in terms of e-reversibility.