International Journal of Mathematics and Mathematical Sciences
Volume 2 (1979), Issue 4, Pages 627-650
doi:10.1155/S0161171279000478
A class of rings which are algebric over the integers
Department of Mathematics, Furman University, Greenville 29613, South Carolina, USA
Received 23 June 1978; Revised 30 October 1978
Copyright © 1979 Douglas F. Rall. 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
A well-known theorem of N. Jacobson states that any periodic associative ring is commutative. Several authors (most notably Kaplansky and Herstein) generalized the “periodic polynomial” condition and were still able to conclude that the rings under consideration were commutative. (See [3]) In this paper we develop a structure theory for a class of rings which properly contains the periodic rings. In particular, an associative ring R is said to be a quasi-anti-integral (QAI) ring if for every a≠0 in R there exist a positive integer k and integers n1,n2,…,nk (all depending on a), so that 0≠n1a=n2a2+…+nkak. In the main theorems of this paper, we show that any QAl-ring is a subdirect sum of prime QAl-rings, which in turn are shown to be left and right orders in division algebras which are algebraic over their prime fields.