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 Marks, G.A. (2006). A criterion for unit-regularity. Acta Math. Hung. 111(4):311–312. DOI: 10.1007/s10474-006-0055-3.[Crossref], [Web of Science ®] , [Google Scholar]] 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.

摘要

元素一个在环- [R是左唯一生成（或左UG）如果，每当太阳神= RB，一个= UB用于单元ü在ř，并且被留下强烈UG如果，每当太阳神= RB，一个= UB = BU为一个单元ü在ř。甲环是左UG（相应地，左强烈UG）环如果它的每个元素是左侧UG（相应的强烈左UG）。受 Marks [ 15]结果的激励 马克斯，GA（2006 年）。单元正则性的标准。数学学报。挂。111(4): 311 – 312。DOI：10.1007/s10474-006-0055-3。[Crossref], [Web of Science®] , [  Google Scholar] ] (von Neumann) 正则环是左UG如果它是单位正则，我们分别建立了正则左UG元素和正则左UG元素的表征强烈的UG元素。作为应用，提出了许多显着的结果，并讨论了与环交换性的联系。