NILPOTENCY OF SUBRINGS GENERATED BY SETS CONTAINED IN THE MIDDLE NUCLEUS

Abstract:Yen and Hentzel [8] studied the nonassociative rings with the ideal generated by setscontained in two of the three nuclei. In this paper we prove that if the additive subgroup S of anonassociative ring R generated by commutators is in the middle nucleus, then (S+SR)R for all positive integers n. Also it is proved that if S+SR = S+RS and S is contained in themiddle nucleus, then the ideal of R generated by S is nilpotent if and only if the subringgenerated by S is nilpotent. We use this to prove that if R is semiprime and the subring generatedby S is nilpotent, then S = 0 and hence R is commutative.