环的交换性定理

本文证明了如下定理:定理1 环R有左单位元,N为R的幂零集元合,(?)x,y∈R,若x≡y((?)od N)就导致x,y与N中元可换或x~k=y~k,x~(k+1)=y~(k+1),其中k=k(x,y)>2,则N为R的理想;且当R/N的每一子环都幂等时,R为交换环.定理2 环R有左单位元且为2-扭自由,N为R的暴零元集合.若V~x,y∈R,x≡y(mod N)就导致x,y与N中元可换或x~k=y~k,x~(k+1)=y~(k+1),k=k(x,y)>2;或x~2=y~2,则N为R的理想,且当R/N的每一子环幂等时,R为交换环.

TWO THEOREMS ON THE COMMUTATIVITY OF RINGS

In this paper, the following theorems were proved:Theorem 1 R is an associative ring with left identity, N is the set of all nilpotents of R. If x = y (mod N)=[x,N] = [y, N] - 0, or there is an integer k=k(x, y)>2 such that xk=yk and xk+1 = yk+1, then N is an ideal of R, and if every sub-ring of R/N is idempotent, then R is commutative.Theorem 2 R is an associative ring with left identy, N is the set of all nilpotent elements of R. If x = y (mod N)=[x, N] = [y, N] = 0, or x2 = y2, or there is an integer k=k(x, y)>2 such that xk=yk and xk+1 = yk+1, when R is 2-torsion free, then N is an ideal of R, and if every sub-ring of R/N is idempotent, then R is commutative.