g(x)-J-clean环

g(x)-J-clean Rings

设R是环,J(R)和C(R)分别表示R的Jacobson根与中心,g(x)∈C(R)[x]为一给定多项式.称R为g(x)-J-clean环,如果任何r∈R可表为r=s+j,其中j∈J(R)且g(s)=0.给出g(x)-J-clean环的基本性质,并给出一些J-clean环的等价刻画,考察(x3-x)-clean环与弱clean环的关系,也证明(xn-1)-J*-clean环就是有限域.

Let R be a ring,J（R） and C（R） denote the Jacobson radical and the center respectively,and g（x） ∈C（R） [x] is a given polynomial. R is called g（x）-J-clean ring,if any r∈R can be written as r = s ＋ j,where j∈J（R） and g（s） = 0. We obtained some basic properties of g（x）-J-clean rings and some equivalent conditions of J-clean rings. We studied the relations between（x3-x）-clean rings and weakly clean rings,and proved that an（xn-1）-J＊-clean ring is a finite field.