On a non-commutative Generalization of Lukasiewicz rings

Abstract

The goal of the present article is to extend the study of commutative rings whose ideals form an MV-algebra as carried out by Belluce and Di Nola to non-commutative rings. We study and characterize all rings whose ideals form a pseudo MV-algebra, which shall be called here generalized Lukasiewicz rings. We obtain that these are (up to isomorphism) exactly the direct sums of unitary special primary rings.