一类伪polar环（英文）

A Class of Pseudopolar Rings

受拟polar环和伪polar环概念的启发,引入根polar环的定义.称环R中的元素α是根polar元,如果存在p^2=p∈R使得p∈comm^2(α),α+p∈U(R)且ap∈J(R);称环R是根polar的,如果R中每个元素都是根polar元.本文研究了根polar环的基本性质并构造了许多例子,同时借助根polar环研究了相关环类.证明了阶数大于1的任意矩阵环都不是根polar的,因此给出局部环上2阶矩阵是根polar元的判定准则.

Motivated by the concepts of quasipolar rings and pseudopolar rings, we intro- duce the notion of radpolar rings. An element a of a ring R is called radpolar if there exists p^2 = p ∈ R such that p ∈ comm^2（a）, a＋p ∈ U（R） and ap ∈ J（R）; R is radpolar if every element of R is radpolar. Basic properties and illustrative examples of this sort of rings are presented; some related rings are studied by means of radpolar rings. It is proved that any matrix ring of size greater than 1 is not radpolar. Consequently, we determine when a 2 × 2 matrix over a local ring is radpolar.