关于半质环交换的条件

ON THE CONDITIONS FOR THE COMMUTATIVITY OF SEMI-PRIME RINGS

本文讨论了一类满足多项恒等式的环的交换性,推广了文[1]的结果,证明了: (1) R为一个结合环,且对任意x,y∈R, a_1xy^2+a_2xyx+a_3x^2y+a_4yx^2+a_5y^2x+a_6yxy∈Z(R) 这里a_1(i=1,2…6)是整数且sum from i=1 to 6(a_1=0),如果下文中条件(Ⅰ,Ⅱ和Ⅲ)之一满足,那么R为交换环。(2) R为一个结合环,对任意x,y_1,y_2…,y_m∈R存在n=n(x,y_1,y_2…,y_m)>1使得: (xy_1y_2…y_m-y_1y_2…y_mx)~n=(xy_1y_2…y_m-y_1y_2…y_mx)。

The main results are the following theorems; Theoeeml Let R be an associative ring, for x, y∈R,a_1xy^2+a_2xyx+a_3x^2y+a_4yx^2+a_6y^2x+a_6yxy∈Z(R),where a_i(i=1, 2, 3, 4, 5, 6) are integers With sum from i=1 to 6 (a_1)=0.R is commutative, if anyone of anyone of(Ⅰ),(Ⅱ);and(Ⅲ) is true,( Ⅰ )a_3+a_4=±2, a_2+a_4=a_2+a_3=(?)1( Ⅱ)a_1+a_5=±2, a_1+a_5=a_5+a_5=(?)1( Ⅲ)a_2+a_6=±2, a_1+a_4=a_3+a_5=(?)1Theorem 2 Let R be an associative ring, for x, y_1, y_2…, y_m∈R and there exist n=(x, y_(1, 2),……, y_m)>1, such that:(xy_1y_2y_3…y_m-y_1y_2y_3…y_mX)~n=(xy_1y_2…y_m-y_1y_2…y_mx), Where m≥1, then R^m(?)Z(R), the center of the ring.