Herstein定理的推广

A Generalization of Herstein's Theorem

1955年Herstein将著名的Jacobson定理推广为:定理A.如果对R中任意x,y,存在可依赖于x,y的整系数多项式p(t),使[x-x2p(x),y]=0,则R是交换的.本文利用多项式的系数和定义了n元多项式,f(x1,x2…,xn)的Fk性质,并以此证明了一个环的交换性定理,当n=1时,即得到定理A.

Herstein generalized Jacobson's famous theorem on the commutativity of rings in 1955 as follows. Theorem A. if for every x, y ∈ R, there is a polynomial p(t) with integer coefficients, depending possibly on x and y, such that [x - x2p(x), y] = 0, then R is commutative. In this paper, we define a property Fk of the polynomial f(x1,X2,…, xn) and prove, on the bassing is property, a theorem on the commutativity of rings. Herstein's theorem is just the one case when n = 1 in our theorem.