具有半交换自同态的环

Rings with Semicommutative Endomorphisms

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珔-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珔-sc环.

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 a--sc,where 〈x^n〉is the ideal by x^n 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 a--sc.