模和环的smal-内射性的一些研究  

Some studies on small-injectivity of modules and rings

作　　者：鲁琦[1] 李娜 LU Qi;LI Na(School of Science,Bengbu University,Bengbu 233030,China)

机构地区：[1]蚌埠学院理学院,安徽蚌埠233030

出　　处：《辽宁师范大学学报（自然科学版）》2020年第3期294-297,共4页Journal of Liaoning Normal University:Natural Science Edition

基　　金：安徽省高校自然科学研究重点项目(KJ2017A569);蚌埠学院自然科学研究重点项目(2017ZR08zd)。

摘　　要：研究了small-内射模和small-内射环的性质,证明了若R是约化的左small-内射环,记S=eRe,e^2=e∈R,则S是约化的左JP-内射环.用单奇异左(右)R-模的small-内射性刻画了半本原环,证明了R是半本原环当且仅当任意单奇异左(右)R-模是small-内射的.得到了在R是半局部环的条件下,以下叙述等价:(1)R是半单环;(2)R是正则环;(3)任意单奇异左(右)R-模是small-内射的;(4)R是半本原环.通过对环的极大左(右)零化子的研究,分别得出了若0≠a∈R,l(a)是R的极大左零化子,则l(a)=l(a^2);若0≠a∈R,r(a)是极大右零化子,则对任意0≠at∈R,有l(a)=l(at),并证得了若R是左small-内射环,且对0≠a∈J,l(a)(r(a))是R的极大左(右)零化子,则a是非零幂零元.Properties of small-injective modules and small-injective rings are studied,it is proved that if R is a reduced small-injective ring,S=eRe,e^2=e∈R,then Sis reduced left JP-injective ring.Small injectivity of simple singular left(right)R-modules is used to characterize semiprimitive rings,it is proved that R is a semiprimitive ring if and only if any simple singular left(right)R-module is small injective.Under the condition that R is semilocal,the following statements are equivalent:(1)R is a simple ring;(2)R is a regular ring;(3)every simple singular left(right)R-module is small injective;(4)R is a semiprimitive ring.Through research of maximal left(right)annihilators of rings,it is respectively given that if 0≠a∈R,l(a)is a maximal left annihilator of R,then l(a)=l(a^2);if 0≠a∈R,r(a)is a maximal right annihilator of R,then for any 0≠at∈R,l(a)=l(at).And it is proved that if R is a small-injective left rings,for 0≠a∈J,l(a)(r(a))is a maximal left(right)annihilator of R,then ais a non-zero nilpotent.

关 键 词：small-内射模 small-内射环 半本原环 

分 类 号：O153.3[理学—数学]