Uniquely Morphic Rings

Abstract

A ring R is called left morphic if, for each a ∈ R, R/Ra ∼ = l(a) or equivalently there exists b ∈ R such that Ra = l(b) and l(a) = Rb, where l(a) and l(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 = a ∈ R there exists a unique b ∈ R such that Ra = l(b) and l(a) = Rb. These rings are completely determined in this paper. We also completely determine the rings R with the property that for each 0 = a ∈ R there exists a unique b ∈ R such that Ra = l(b).