半正则环的几点注记

通过GP-内射性和small内射性研究环的半本原性和正则性,证明了在J(R)是约化的条件下,如下条件等价:(1)R是正则环;(2)R是半正则环且对J(R)的每个元a,存在正整数n,使得Ran是GP-内射模;(3)R是半正则环且每个单奇异的左R-模都是small内射模;(4)R是半正则环且对J(R)的每个元a,存在正整数n,使得Ran是EP-内射模。

Some Remarks on Semiregular Rings

In this paper,we mainly study the semiprimitivity and regularity of semiregular rings by GP-injective modules and small-injective modules,and prove that if J(R) is reduced,the following conditions are equivalent:(1)R is a von Neumann regular ring;(2)Ris a semiregular ring and for every a∈J(R),there exists a positive integer n such that Ran is GP-injective;(3)R is a semiregular ring and every simple singular left R-module is-injective;(4)R is a semiregular ring and for every a∈J(R),there exists a positive integer such that Ran is EP-injective.