On a characterization of distributive rings via saturations and its applications to Armendariz and Gaussian rings

Abstract

In this paper we apply Ferrero–Sant’Ana’s characterization of right distributive rings via saturations to prove that all right distributive rings are Armendariz relative to any unique product monoid. As an immediate consequence we obtain that all right distributive rings are Armendariz. We apply this result to give a new proof of the well-known fact that all right duo right distributive rings are right Gaussian. We also show that for any nontrivial unique product monoid $S$, the class of Armendariz rings relative to $S$ is contained in the class of Armendariz rings, and we present an example of a unique product monoid $S$ for which this containment is strict.