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 n 1 , n 2 , … , n k (all depending on a ), so that 0 ≠ n 1 a = n 2 a 2 + … + n k a k . 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.