摘要:
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 semi - prime rings. If (A) x1,x2,… ∈R ,there exist a polyno mial p(t) with integer coefficients ,containing such that 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 wordfx(x,y) containing x2 and n = n(a,b) (≥K)y s,and a polynomial Φx(x,y) with integer coefficients such that […[[k∑I=0αibiabk-ifx(a,b),x2],x3],…,xnZ(R)then R is commutative.%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(v)x1,x2,…,xn∈R,有依赖于x1,x2的整系数多项式P(t)使得[…[[x1-x21p(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(v)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Ψ(x,y)使得[…[[kΣi=0aibiabk-1-fx(a,b)Ψx(a,b),x2],x3],…,xn]∈Z(R)其中|k∑αi|=1,则R为交换环.