Order in a special class of rings and a structure theorem

Abstract:

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