On a non-commutative Generalization of Lukasiewicz rings

TL;DR

This paper extends the concept of MV-algebraic ideal structures from commutative rings to non-commutative rings, characterizing generalized Lukasiewicz rings as direct sums of unitary special primary rings.

Contribution

It introduces and characterizes generalized Lukasiewicz rings, expanding the algebraic framework from commutative to non-commutative rings with pseudo MV-algebraic ideals.

Findings

Generalized Lukasiewicz rings are exactly the direct sums of unitary special primary rings.

The study extends MV-algebraic ideal structures to non-commutative rings.

Provides a complete characterization of these rings up to isomorphism.

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.