带有自同态的J-quasipolar矩阵环（英文）

On J-quasipolar Matrix Rings with Endomorphisms

对于环R中的一个元素a,如果存在p^2=p∈comm^2(a)使得a+p∈J(R),则称a为J-quasipolar的,一个环称为J-quasipolar的如果环中每一个元素都是J-quasipolar的.本文中我们研究了带有自同态的3×3阶矩阵环T_3(R;σ)的J-quasipolar性质.设R是一个局部环,σ:R→R是环R的自同态,如果σ(J(R))?J(R),我们证明了T_3(R;σ)是J-quasipolar的当且仅当R是唯一bleached环的并且R/J(R)??2.

An element aof a ring Ris J-quasipolar if there exists p2=p∈comm2(a)such that a+p∈J(R),where J(R)is the Jacobson radical of R.A ring is called J-quasipolar if every element is J-quasipolar.In this paper,the property of 3×3 matrix T3(R;σ)with endomorphisms is investigated.Let Rbe a local ring andσ:R→Rbe an endomorphism of R.Ifσ(J(R))?J(R),it is proved that T3(R;σ)is J-quasipolar if and only if Ris a uniquely bleached ring and R/J(R)■Z2.