Annihilator-stability and unique generation.

A ring R is said to be left uniquely generated if R a = R b in R implies that a = u b 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 | r b = 0 } , a theorem of Canfell asserts that R is left uniquely generated if and only if, whenever R a + 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. [ABSTRACT FROM AUTHOR]