半素环的一个交换性定理

A Commutativity Theorem on Semiprime Rings

证明了半素环Ｒ为交换的，如果它满足条件（ａｋ）：对任意ａ，ｂ∈Ｒ，存在一个字长＞Ｋ且含有（ｘｙ）＾２（或（ｙｘ）×２）的字Ｗｘ（ｘ，ｙ）及一个能被Ｗｘ（ｘ，ｙ）整除的整系数多项式ｆｘ（ｘ，ｙ），使得ａｂ＾ｋ－ｆｘ（ａ，ｂ）∈Ｚ（Ｒ），其中Ｋ是一个给定的正整数，Ｗｘ（ｘ，ｙ）与ｆｘ（ｘ，ｙ）均可随Ｘ＝（ａ，ｂ）而变，Ｚ（Ｒ）是环Ｒ的中心。

Proves that a semiprime rings R with centre Z(R) is commutative if it satisfies the condition(αk):for arbitrary a,b ∈ R,there exist a word W_x(x,y) containing (xy)￣2(or(yx)￣2) and its length>K and a polynomial fx(x,y) with integer coefficients which can be divided exactly by W_x(x,y) such that ab￣K-fx(a,b) ∈ Z(R),where K is a fixed positive integer,W_x(x,y) and f_x(x,y) can vary as X=(a,b)varies.