International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 2, Pages 403-406
doi:10.1155/S0161171284000417
Periodic rings with commuting nilpotents
1Department of Mathematics, University of Petroleum and Minerals, Dhahran, Saudi Arabia
2Department of Mathematics, University of California, Santa Barbara 93106, California, USA
Received 16 August 1983
Copyright © 1984 Hazar Abu-Khuzam and Adil Yaqub. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let R be a ring (not necessarily with identity) and let N denote the set of nilpotent elements of R. Suppose that (i) N is commutative, (ii) for every x in R, there exists a positive integer k=k(x) and a polynomial f(λ)=fx(λ) with integer coefficients such that xk=xk+1f(x), (iii) the set In={x|xn=x} where n is a fixed integer, n>1, is an ideal in R. Then R is a subdirect sum of finite fields of at most n elements and a nil commutative ring. This theorem, generalizes the “xn=x” theorem of Jacobson, and (taking n=2) also yields the well known structure of a Boolean ring. An Example is given which shows that this theorem need not be true if we merely assume that In is a subring of R.