Kthe半单纯环的几个交换性定理

SOME COMMUTATIVITY THEOREM OF KOTHE SEMISIMPLE RINGS

设 R 是一个 Kthe 半单纯环,C 是 R 的中心.本文证明,R 满足下列条件之一时为交换环:(1)对任意 x,y∈R,存在自然数 l=l(x,y),m=m(x,y)>1,n=n(x,y),且 l≤n,使得下列关系式之一恒成立:(i)xy^l-x^my^n∈C;(ii)xy^l-y^nx^m∈C;(iii)x^ly-x^ny^m∈C;(iv)x^ly-y^mx^n∈C.(2)R 不含非零的诣零单边理想,且对任意 x,y∈R,存在自然数 l=l(y,y)>1,n==n(x,y),n≥l,使得下列关系式之一恒成立:(i)xy^l-(xy)~n∈C;(ii)xy^l-(yx)~n∈C;(iii)x^ly-(xy)~n∈C;(iv)x^ey-(yx)~n∈C.

In this paper we proved:let R be a kothe semisimple ring,and C be thecentre of R,then R is commutative if R satisfis one of the following con diti-ons:(1)for every x,y∈R,there exists natural numbers l=l(x,y),m=m(x,y)>1 n=n(x,y) and n≥1 such that one of the following relation always ho-ld:(i)xy^l-x^my^n∈C;(ii)xy^l-y^nx^m∈C;(iii)x^ly-x^my^n∈C;(iV)x^ly-y^mx^n∈C;2)R has no non—zero nil one—side ideas,and for every x,y∈R,ther-e exists natural numbers l=l(x,y)>1,n=n(x,y)≥l such that one of the fo-llowing relation always holds:(i)xy)~l-(xy)~n∈C; (ii)xy^l-(yx)~n∈C;(iii)x^ly-(xy)~n∈C; (iV)x^ly-(yx)~n∈C.