左α-半交换环及其扩张

Left α-Semicommutative Rings and Their Extensions

本文通过引入左α-半交换环推广半交换环的概念。设α是环R的一个非零自同态,称R是一个左α-半交换环,如果对任何a,b∈R,由ab=0可以推出α(a)Rb=0。本文讨论左α-半交换环与相关环的关系,给出左α-半交换环的一些扩张性质,证明了:①环R是α-rigid环当且仅当R是约化的左α-半交换环,且α是单同态;②如果R是约化的左α-半交换环,则R[x]/〈xn〉是左珔α-半交换环,其中〈xn〉是由xn生成的理想,n为任何正整数。

The notion of semicommutative rings is extended by introducing left α-xemicommutative rings. The relationship between left α-semicommutative rings and related rings are discussed and some extensions of left α-semicommutative rings are investigated. It is shown that 1） a ring R is α-rigid if and only if R is a reduced left α-semicommutative ring, and a is a monomorphism. 2） If R is a reduced left α-semicommutative ring, then R[x]/〈xn〉 is a left α-semicommutative ring, where （x^n） denotes the ideal generated by xn and n is any positive integer.