A class of rings which are algebric over the integers

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.