Commutative MTL-rings

TL;DR

This paper introduces MTL-rings, a class of commutative rings with ideal lattices forming MTL-algebras, and characterizes their structure, especially in non-noetherian cases like valuation and Pr"ufer domains.

Contribution

It characterizes MTL-rings in local and noetherian cases and constructs examples from valuation and Pr"ufer domains, expanding understanding of their algebraic properties.

Findings

Local MTL-rings are exactly arithmetical rings.

Noetherian MTL-rings are BL-rings.

Non-noetherian valuation rings are primary examples of MTL-rings.

Abstract

We introduce in this work, the class of commutative rings whose lattice of ideals forms an MTL-algebra which is not necessary a BL-algebra. The so-called class of rings will be named MTL-rings. We prove that a local commutative ring with identity is an MTL-ring if and only if it is an arithmetical ring. It is shown that a noetherian commutative ring R with an identity is an MTL-ring if and only if ideals of the localization RM at a maximal ideal M are totally ordered by the set inclusion. Remarking that noetherian MTL-rings are again BL-rings, we work outside of the noetherian case by considering non-noetherian valuation domains and non-noetherian Pr\"ufer domains. We established that non-noetherian valuation rings are the main examples of MTL-rings which are not BL-rings. This leads us to some constructions of MTL-rings from Pr\"ufer domains: the case of holomorphic functions ring through their algebraic properties and the case of semilocal Pr\"ufer domains through the theorem of independency of valuations. We end up giving a representation of MTL-rings in terms of subdirectly irreducible product.