ZC—环与Von Neumann正则环

本文将已有的一些可换环的结论推广到一类非可换环上去,同时还改进了某些结论,得到了如下主要结果: 设A是零因子可换环,那么以下条件等价: (1)A是正则环; (2)A是V-环且A的每个极大本质左理想是双边; (3)每个单奇异A-模是P-内射的,且A的每个极大本质左理想是双边的; (4)A的每个极大本质左理想是P-内射的; (5)A的每个本质左零化子是P-内射的; (6)存在忠实左A-模C使当d∈C且l(d)是本质的时,l(d)是P-内射的; (7)A中每个主左理想是平坦左零化子, (8)A包含极大左理想五使当k∈K且,l(K)是本质的时,l(k)是P-内射的。

ZC-Rings and Von Neumann Regular Rings

In this note, we generalise some results of commutatlve. rings to a class of noneommutative rings. And the results of some theorems are. improved. We prove the following results: Let A be a zero divisors comrnutative ring, then the following are equivalent: (1) A is a Von Neumann regular ring; (2) A is a V-ring and every maximal essential left ideal of: A is twosided; (3) Every simple singular A-module is p-injective, and every maximal essential left ideal of A is two-sided; (4) Every maximal essential left ideal of A is p-injective; (5) Every essential left annihilator of an element of A is p-injeetive; (6) There exists a faithful left A-module C such that every essential left annihilator of an element of C is p-injective; (7) Every principal left ideal of A is a flat annihtlator; (8) A contains a maximal left ideal K such that every essential annihilator of an element of K is p-injective.