Annihilator-stability and unique generation

A ring R is said to be left uniquely generated if Ra=Rb in R implies that a=ub for some unit u in R. These rings have been of interest since Kaplansky introduced them in 1949 in his classic study of elementary divisors. Writing l(b)={r∈R|rb=0}, a theorem of Canfell asserts that R is left uniquely generated if and only if, whenever Ra+l(b)=R where a,b∈R, then a−u∈l(b) for some unit u in R. By analogy with the stable range 1 condition we call a ring with this property left annihilator-stable. In this paper we exploit this perspective on the left UG rings to construct new examples and derive new results. For example, writing J for the Jacobson radical, we show that a semiregular ring R is left annihilator-stable if and only if R/J is unit-regular, an analogue of Bass' theorem that semilocal rings have stable range 1.