ZWGP-内射性与环的非奇异性

ZWGP-injectivity and nonsingularity of rings

引入了ZWGP-内射模和ZWGP-内射环的概念,对ZWGP-内射环进行了等价刻画。研究了ZWGP-内射模(环)的性质,举例说明了ZWGP-内射环和非奇异环的关系。给出了环是非奇异的充分必要条件。证明了:(1)若环R的左零化子是R的W-理想,且R的任意本质理想均是左ZWGP-内射的,则R是左非奇异环;(2)若对R的任意本质左理想I,R/I是ZWGP-内射的,且l(a_1)■l(a_1a_2)■l(a_1a-2a_3)■…是平稳的,ai∈Z(_RR),i=1,2,3,…,则R是左非奇异的。

ZWGP-injective modules and ZWGP-injective rings are introduced; equivalent characterizations of ZWGP-injective rings are obtained. The properties of ZWGP-injective modules（ rings） are explored,the relations between ZWGP-injective rings and nonsingular rings are illustrated by some examples,and sufficient and necessary conditions that rings are nonsingular are given. It is proved that：（ 1） If left annihilators of R are W-ideals of R,and any essential ideals of R are left ZWGP-injective,then R is left nonsingular;（ 2） If for any essential left ideal I,R / I is ZWGP-injective,and l（ a_1）■l（ a_1a_2）■l（ a_1a_2a_3）■… is stable,ai∈Z（_RR）,i = 1,2,3,…,then R is left nonsingular.