半素环的交换性

讨论元素满足两个以上多项式关系之一的半素环的交换性,证明了:定理1 R为半素环,(?)x,y∈R,若x,y满足如下3个关系式之一,则R为交换环:(i)(xy)~m-(xy)~(m_1)(yx)~(m_2)∈Z(R);(ii)(xy)~5-(yx)~1∈Z(R);(iii)(xy)~(k_1)(yx)~(k_2)-(yx)~(k_2)(xy)~(k_1)∈Z(R).其中m,m_i,k_i,s及t与x,y有关且m_1+m_2,t,k_1+k_2为有界自然数.定理2 R为半素环,若R满足下述四个条件之一,则R可换:(1)(?)x,y∈R,x~(2m)y~(2n)-x~my~(2n)x~m∈Z(R)或x~sy~t-y~tx~s∈Z(R);(2)(?)x,y∈R,x~(2m)y~(2n)-y~nx~(2m)y~n∈Z(R)或x~sy~t-y~tx~s∈Z(R);(3)(?)x,y∈R,(yx)~n-yx~ny~(n-1)∈Z(R)或(xy)~n-x~ny~n∈Z(R);(4)(?)x,y∈R,(yx)~n-x~(n-1)y~nx∈Z(R)或(xy)~n-x~ny~n∈Z(R).其中m,n,s,t为自然数,而(1)及(2)中的m,n,s,t与x,y相关,(3)及(4)中n(>1)只与x(或y)有关.

SOME CONDITIONS FOR COMMUTATIVITY OF SEMIPRIME RINGS

In the paper, the following theorems arc proved:Theorem 1 Let R be a scmiprimc ring with center Z(R), for all x,y(- R, if x,y satisfy one of the following three conditions, then R is commutative:(i) (xy)m-(xy)m1(yx)"Z(R);(ii) (xy)'-(yx)'Z(R);(iii) (xy)k1 (yx)k2 - (yx)k2 (xy)k1Z(R).Where the natural number m, mt, kt, s and t arc dependent on x and y, and k1+k2 and / arc bounded.Theorem 2 Lcl R be a scmiprimc ring, if R satisfies one of the following four conditions ,thcn R is commutative:(1) V x.y 6 R, cither x," y2" - x" yx" eZ(R)orx' y' -y'x'eZ(R);(2) Vx,yR,cilbcrx2my2' - y" x," y" eZ(R)orx' y' -y'x'eZ(R);(3) V x.y 6 R, cither (yx)"-yxn yn-' 6 Z(R) or (xy)"-xnyn 6 Z(R);(4) V x,y 6 R, cither (yx)"-, 2(R) or (xy),-xay" Z(R).In (1) and (2), the natural number m,n,s and / arc dependent on x and y, and they arc bounded; in (3) and (4), n > 1 is only dependent on x (or y ).