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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 [Citation15] 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.
2020 MATHEMATICS SUBJECT CLASSIFICATION:
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Funding
This research was supported by a Discovery Grant (grant number RGPIN-2016-04706) from NSERC of Canada.