具有半交换自同态的环

通过引入半交换自同态的概念, 研究具有半交换自同态的环(简称α-sc环). 对任何a,b∈R, 如果α(a)b=0, 有aRα(b)=0, 则环R的一个自同态α称为半交换的.
给出α-sc环与相关环的关系及α-sc环的一些扩张性质, 证明了： 1) 设α是约化环R的自同态, 则R是α-sc]环当且仅当R［x］/〈xⁿ〉是α-sc环, 其中〈xⁿ〉是由xⁿ生成的理想, n为任何正整数; 2) 设α是环R的自同构, R是对称的右Ore环, 则R是α-sc环当且仅当R的经典右商环Q(R)是α-sc环.

Rings with Semicommutative Endomorphisms

We investigated the rings with semicommutative endomorphisms, referred to as the α-sc ring, by means of introducing the notion of semicommutative endomorphisms. A endomorphism α of a ring R is called semicommutative if α(a)b=0 implies aRα(b)=0 for any a,b∈R. We discussed the relations between α sc rings and related rings and present some extensions of α sc rings. It is proved
that: 1) if α is a endomorphism of a reduced ring R, then R is α sc if and only if R［x］/〈xⁿ〉 is α-sc, where 〈xⁿ〉is the ideal by xⁿ generated in R and n is any positive integer; 2) if α is an automorphism of R, R is a symmetric right Ore ring, then R is α sc if and only if the classical right quotient ring Q(R) of R is α-sc.