2021/06/19 by Samuel Mouchili, Surdive Atamewoue, Mouchili, Samuel +7
Chemistry · Computer Science · Decision Sciences · Mathematics · #Advanced Algebra and Logic #Algebra over a field #Chemistry #Combinatorics #Commutative Algebra (math.AC) #Commutative algebra #Commutative property #Commutative ring #Discrete mathematics #FOS: Mathematics #Fuzzy and Soft Set Theory #Local ring #Mathematics #Noetherian #Noetherian ring #Noncommutative ring #Prime (order theory) #Pure mathematics #Ring (chemistry) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #Semiprime ring #Von Neumann regular ring #math.AC #math.RA
paper · pdf · doi:10.48550/arxiv.2106.10428
published in arXiv (Cornell University) (Cornell University)
arxiv created 2021/06/19 · openalex publication_date 2021/06/19 · arxiv updated 2021/06/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 Prüfer 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 Prüfer domains: the case of holomorphic functions ring through their algebraic properties and the case of semilocal Prüfer domains through the theorem of independency of valuations. We end up giving a representation of MTL-rings in terms of subdirectly irreducible product.