关于环交换性的两个定理

Two commutative theorem of rings

为了促进交换性的发展,根据半质环及半单环的相关资料,扩展了文献[1-2]的结论,得出了环的两个交换性定理:定理1:设R为一个半质环,若对▽x1,x2,…,xn∈R,有依赖于x1,x2的整系数多项式p(t)使得[…[[x1-x12p(x1),x2],x3],…,xn]∈Z(R),则R为交换环。定理2:设R为一个kothe半单纯环,若对▽a,b,x2,…,xn∈R都有一正整数K=K(a,b),一含有x2和n=n(a,b)(≥K)个y的字fx(x,y)及一整系数多项式φx(x,y)使得[…[[∑ki=0αi bi abk-i-fx(a,b)φx(a,b),x2],x3],…,xn]∈Z(R)其中|∑ki=0αi|=1,则R为交换环.

For the rapid development of commutative, two commutative theorems of rings were given, with the results of the improvement of semi - prime rings and kothe - semisimple rings ： Theoreml ：Let R is a semiprime rings. If arbitary x1 ,x2,… ,xn ∈ R ,there exist a polynomial p （t） with integer coefficients , containing such that […[[x1-x1^2p（x1）,x2],x3],…,xn]∈Z（R）then R is a commutative. Theorem2 ：If R is a kothe - semisimple rings and for arbitrary a, b,x2,… ,xn∈R, there exist a positive integer K=K（a,b）, a word fx（x,y） containing x^2 and n = n（a,b）（≥K） s, and a polynomial φx （x,y） with integer coefficients such that […[[∑i=0^k αib^iab^k-1-fx（a,b）αx（a,b）,x2],x3],…,xn]∈Z（R） ｜∑i=0^k αi｜=1,then R is commutative.