Reversible ring property via idempotent elements

Abstract

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.