Left uniquely generated elements in rings. Issue 9 (2nd September 2021)

• Record Type: Journal Article
• Authors: Zhou, Yiqiang
• Is Part Of: Communications in algebra. Volume 49:Issue 9(2021)
• Page Start: 3825
• Page End: 3836
• Publication Date: 2021-09-02
• Subjects: Commutativity -- (strongly) UG element -- (strongly) UG ring -- unit -- (unit-) regular ring; Primary 16U60 -- 16E50 -- 16U99; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2021.1908549
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: An element a in a ring R is left uniquely generated (or left UG ) if, whenever Ra = Rb, a = ub for a unit u in R, and is left strongly UG if, whenever Ra = Rb, a = ub = bu for a unit u in R . A ring is a left UG (resp. left strongly UG ) ring if each of its elements is left UG (resp. left strongly UG ). Motivated by a result of Marks [15 ] that a (von Neumann) regular ring is left UG iff it is unit-regular, we establish characterizations of a regular left UG element and, respectively, a regular left strongly UG element. As applications, many notable consequences are presented and connections with commutativity of rings are discussed.