Order in a special class of rings and a structure theorem

Abstract

Below a special class of not necessarily associative or commutative rings A A is considered which is characterized by the property that A A has no nonzero nilpotent element and that a product of elements of A A which is equal to zero remains equal to zero no matter how its factors are associated. It is shown that ( A , ⩽ ) (A, \leqslant ) is a partially ordered set where x ⩽ y x \leqslant y if and only if x y = x 2 xy = {x^2} . Also it is shown that ( A , ⩽ ) (A, \leqslant ) is infinitely distributive, i.e., r sup x i = sup r x i r\sup {x_i} = \sup r{x_i} . Finally, based on Zorn’s lemma it is shown that A A is isomorphic to a subdirect product of not necessarily associative or commutative rings without zero divisors.