A non commutative generalization BL-rings

TL;DR

This paper extends the concept of BL-algebra structures on ideals from commutative rings to non-commutative rings, introducing pseudo BL-rings and characterizing their structure.

Contribution

It introduces pseudo BL-rings as a non-commutative generalization and characterizes their structure via subdirectly irreducible factors.

Findings

Ideals of pseudo BL-rings form a pseudo BL-algebra.

Pseudo BL-rings are subrings of direct sums of specific primary and valuation rings.

The structure of pseudo BL-rings is fully characterized in terms of these subdirect factors.

Abstract

The purpose of this work is to extend the study of the commutative rings whose lattice of ideals can be a structure of BL-algebra as carry out by Heubo et al in 2018, to non commutative rings appointed in the work as pseudo BL-rings. We study and characterize rings whose ideals form a pseudo BL-algebra, we describe them in terms of their subdirectly irreductible factors. We obtain that these are (up to isomorphism) to a subring of a direct sums of unitary special primary rings and discrete valuation ring.