Rings in which Every Element Is A Left Zero-Divisor

DOI：10.3770/j.issn:2095-2651.2013.04.003

 作者 单位 任艳丽 南京晓庄学院数学与信息技术学院, 江苏 南京 211171 王尧 南京信息工程大学数学与统计学院, 江苏 南京 210044

本文引进左(右)零因子环的概念,它们是一类无单位元的环.我们称一个环为左(右)零因子环,如果对于任何 $a \in R$,都有$r_R (a) \neq 0~(l_R(a)\neq 0)$,而称一个环为强左(右)零因子环,如果$r_R(R)\neq 0~(l_R(R)\neq 0)$.Camillo和Nielson称一个环$R$为右有限零化环(简称RFA-环),如果$R$的每一个有限子集都有非零的右零化子.本文给出左零因子环的一些基本例子,探讨强左零因子环和RFA-环的扩张,并给出它们的等价刻画.

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.