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arXiv · 2106.10428

Commutative MTL-rings

Samuel Mouchili·Surdive Atamewoue·Selestin Ndjeya·Olivier Heubo-Kwegna
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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.

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Samuel Mouchili, Surdive Atamewoue, Selestin Ndjeya, Olivier Heubo-Kwegna. 2021-06-19. Commutative MTL-rings. https://arxiv.org/abs/2106.10428

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