UNIQUELY MORPHIC RINGS

Abstract

A ring R is called left morphic if, for each a is an element of R, R/Ra congruent to 1(a) or equivalently there exists b is an element of R such that Ra = 1(b) and 1(a) = Rb, where 1(a) and 1(b) denote the left annihilators of a and b in R, respectively. Motivated by recent work on left morphic rings, we study the rings R satisfying the property that for each 0 not equal a is an element of R there exists a unique b is an element of R such that Ra = 1(b) and 1(a) = Rb. These rings are completely determined in this paper. We also completely determine the rings R with the property that for each 0 not equal a is an element of R there exists a unique b is an element of R such that Ra = 1(b).