A non-commutative generalization of Łukasiewicz 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 [1] 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.

A non commutative generalization BL-rings

“…The natural question that arises is what happens if one drops the commutativity assumption on Lukasiewicz rings. The answer of that question has been the goal of Kadji, Lele and Nganou in their work A non-commutative generalization of Lukasiewicz rings [7] where they study and characterize all rings whose ideals form a pseudo MV-algebra, which they called generalized Lukasiewicz rings, this was in 2016. Since the class of BL-algebras contains the MV-algebras, from then in 2018, Heubo et al [8] initiated the study of commutative rings whose lattice of ideals can be equipped with a structure of BL-algebra.…”