Rings in which Every Element Is A Left Zero-Divisor

We introduce the concepts of left (right) zero-divisor rings, a class of rings without identity. We call a ring R left (right) zero-divisor if rR (a) = 0 (lR (a) = 0) for every a∈R, and call R strong left (right) zero-divisor if rR(R) = 0 (lR(R) = 0). Camillo and Nielson called a ring right finite annihilated (RFA) if every finite subset has non-zero right annihilator. We present in this paper some basic examples of left zero-divisor rings, and investigate the extensions of strong left zero-divisor rings and RFA rings, giving their equivalent characterizations.