Rings in which Every Element Is A Left Zero-Divisor
Received:May 10, 2012  Revised:November 22, 2012
Key Word: zero-divisor   left zero-divisor ring   strong left zero-divisor ring   RFA ring   extensions of rings.
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant Nos.11071097; 11101217).
 Author Name Affiliation Yanli REN School of Mathematics and Information Technology, Nanjing Xiaozhuang University, Jiangsu 211171, P. R. China Yao WANG School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Jiangsu 210044, P. R. China
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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.