%0 Journal Article
%T A class of rings which are algebric over the integers
%A Douglas F. Rall
%J International Journal of Mathematics and Mathematical Sciences
%D 1979
%I Hindawi Publishing Corporation
%R 10.1155/s0161171279000478
%X 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 ”é ”ė  0 in R there exist a positive integer k and integers n1,n2, ”é ? |,nk (all depending on a), so that 0 ”é ”ė  n1a=n2a2+ ”é ? |+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.
%K anti-integral
%K quasi-anti-integral
%K periodic
%K prime.
%U http://dx.doi.org/10.1155/S0161171279000478