International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 2, Pages 403-406
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.
Let be a ring (not necessarily with identity) and let denote the set of nilpotent elements of . Suppose that (i) is commutative, (ii) for every in , there exists a positive integer and a polynomial with integer coefficients such that , (iii) the set where is a fixed integer, , is an ideal in . Then is a subdirect sum of finite fields of at most elements and a nil commutative ring. This theorem, generalizes the theorem of Jacobson, and (taking ) 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 is a subring of .