拟对偶双边模与对偶环  

Quasi-dual Bimodules and Dual Rings

作　　者：李爱华[1] 黎奇升[1] 

机构地区：[1]吉首大学数学与计算机科学系,湖南吉首416000

出　　处：《吉首大学学报》2001年第4期86-88,共3页

基　　金：湖南省教育厅自然科学基金资助项目 (99C73)

摘　　要：左拟对偶双边模 SMR 可以被刻划成MR 的任意子模K和SS的任意左理想L分别是rMlS(K)和lSrM(L)的一个直和项 .对一个左拟对偶双边模 SMR,有以下结论 :(1) SM为Kasch模 ;(2 )rMlS(Soc(MR) ) =Soc(MR) ,lSrM(Soc(SS) ) =Soc(SS) ;(3)lS(Soc(MR) ) J(S) ,rM(Soc(SS) ) Rad(MR) ;(4 )若MR 为CS -模 ,则Soc(MR) eMR;(5 )若MR 是非M -奇异的 ,则M是半单的 ;(6 )若MR 在σ[M]中投射且MR 半单 ,则M是非M -奇异模 .并且还得出 ,若R是左对偶环或左拟对偶环 ,则R是半单环当且仅当R非奇异 .In this paper, left quasi-dual bimodules can be characterized as those bimodules for which any submodule K of M R and any left idea L of SS are a direct summand of r Ml S (K) and l Sr M (L) respectly. For a left quasi-dual bimodule S M R, some results are proved:(1) SM is a Kasch module,(2)r Ml SWTB(Soc(M R))=Soc(M R) and l Sr M(Soc(X] SS))=Soc( SS),(3)l S(Soc(M R))J(S) and r M(Soc( SSWTB))Rad(M R),(4) if M R is a CS-module, the Soc(M R) eM R, (5)if M R is nonsingular, then M is semisimple,(6) if M R is projective in σ SS))=Soc( SS),(3)l S(Soc(M R))J(S) and r M(Soc( SSWTB))Rad(M R),(4) if M R is a CS-module, the Soc(M R) eM R, (5)if M R is nonsingular, then M is semisimple,(6) if M R is projective in σ and M R is semisimple, the M is a nonsingular module. It deduced that if R is a left dual ring or a left quasi-dual ring, then R is a semisimple if R is nonsingular.

关 键 词：拟对偶双边模 M-奇异模 Kasch模 对偶环 半单环 结合环 本质左理想 

分 类 号：O153.3[理学—数学]