Left uniquely generated elements in rings. Issue 9 (2nd September 2021) - Zhou, Yiqiang
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.