小内射环的扩张  

Extensions of small injective rings

作　　者：沈亮[1] 陈建龙[1] 

机构地区：[1]东南大学数学系,南京210096

出　　处：《东南大学学报（自然科学版）》2013年第6期1340-1342,共3页Journal of Southeast University：Natural Science Edition

基　　金：江苏省自然科学基金资助项目(BK20130599;BK2010393);教育部留学回国人员科研启动基金资助项目;国家教育部博士点专项基金资助项目(20120092110020)

摘　　要：设R为环,R的右理想I称为小理想如果对任意R的真右理想K都有I+K≠R.环R称为右小内射环如果每个从R的小右理想I到R R的同态可扩张为从R R到R R的同态.左小内射环定义类似.讨论了环的扩张如平凡扩张、形式三角矩阵环、上三角矩阵环等的小内射性.证明了环R通过双模R V R的平凡扩张S=R∝V为右自内射环当且仅当S为右小内射环当且仅当V作为右R-模为自内射模且R=End V R.并证明了非平凡的上三角矩阵环一定不是右小内射环.Let R be a ring.A right ideal I of R is called small if,for every proper right ideal K of R,I ＋K≠R.A ring R is called right small injective if every homomorphism from a small right ideal I of R to RR can be extended to an R-homomorphism from RR to RR.Left small injective rings can be defined similarly.Small injectivities of some extensions of rings such as trivial extensions,formal triangular matrix rings,upper triangular matrix rings and so on are discussed.It is proved that the trivial extension S =R∝V of R by the bimodule R VR is right self-injective if and only if S is right small injective if and only if V is self-injective as a right R-module and R =End VR.It is also shown that any nontrivial upper triangular matrix ring cannot be right small injective.

关 键 词：小内射环 环的平凡扩张 形式三角矩阵环 上三角矩阵环 

分 类 号：O153.3[理学—数学]