BL-rings

Heubo-Kwegna, Olivier A.; Lele, Celestin; Ndjeya, Selestin; Nganou, Jean B.

doi:10.1093/jigpal/jzy001

Cited by 6 publications

(12 citation statements)

References 9 publications

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“…Since R/K x is a pseudo BL-ring by Corollary 3.16.2 and Proposition 3.20. Let pick up the set{R x | x ∈ R \ {0}} to be {R/K x | x ∈ R \ {0}}.Then the rest of the proof is similar to Theorem 4.2 in[8].The following corollary is the same as Corollary 4.3 in[8].Corollary 4.7.1 Let R be a subdirectly irreductible BL-ring. Then (1) For every annihilator ideal I = R and every dense ideal J with I ⊆ J, then J → I = I and J I = I.…”

mentioning

confidence: 88%

“…So R = {0} contradiction). Thus by Proposition [2.10, [8]] there exists a prime ideal Q of R such that (xR) * ⊆ Q and (Rx) − ⊆ Q.…”

Section: Definition 323 ([8]mentioning

confidence: 99%

“…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. Recall that BL-algebras where introduced by Peter Hájek in 1998 as algebraic structures for his Basic Logic to formalize manyvalued logics (in short MV logics) induced by continuous t-norms on the real unit interval [0,1] and the were generalized to pseudo BL-algebras by Di Nola, Georgescu and Iorgulescu [3].…”

Section: Introductionmentioning

confidence: 99%

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A non commutative generalization BL-rings

Tsafack¹,

Fobasso²,

Feng³

et al. 2021

Preprint

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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.

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mentioning

confidence: 88%

“…So R = {0} contradiction). Thus by Proposition [2.10, [8]] there exists a prime ideal Q of R such that (xR) * ⊆ Q and (Rx) − ⊆ Q.…”

Section: Definition 323 ([8]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A non commutative generalization BL-rings

Tsafack¹,

Fobasso²,

Feng³

et al. 2021

Preprint

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“…Since rings are very important algebraic tools in mathematics which have very good and nice properties, many mathematicians have started to relate those algebras with the well known classes of rings. We have for example Jensen for Arithmetical rings [13], Belluce, Di Nola and Marchioni for MV-rings and Godelrings [2,3], Heubo-Kwegna, Lele, Ndjeya and Nganou for BL-rings [10].…”

Section: Introductionmentioning

confidence: 99%

“…One year later (in 2010), the ring structure for Gödel algebras known as Gödel rings were introduced ( [3]). Recently in 2018, BL-rings were presented in [10] as rings whose ideals form a BL-algebra. The natural question that one can ask is the following.…”

Section: Introductionmentioning

confidence: 99%

Commutative MTL-rings

Mouchili¹,

Atamewoue²,

Ndjeya³

et al. 2021

Preprint

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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 R M 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.

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Rings and residuated lattices whose fuzzy ideals form a Boolean algebra

Foka

Tonga

2021

Soft Comput

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BL-rings

Cited by 6 publications

References 9 publications

A non commutative generalization BL-rings

A non commutative generalization BL-rings

Commutative MTL-rings

Rings and residuated lattices whose fuzzy ideals form a Boolean algebra

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