Rings in which Every Element Is A Left Zero-Divisor

Abstract:

      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 $r_{R}(a) \neq 0~(l_{R}(a) \neq 0)$ for every $a\in R$, and call $R$ strong left (right) zero-divisor if $r _{R} (R) \neq 0$~($l_{R}(R) \neq 0$). 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.