Two-sided residuation in the set of topologizing filters on a ring

Arega, Nega; Berg, John van den

doi:10.1080/00927872.2016.1172624

Cited by 2 publications

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“…This section and the next, provide the torsion theoretic background that is necessary for what follows. A slightly more detailed exposition may be found in the early pages of [1]. For further background, we refer the reader to the texts [6], [7] and [10].…”

Section: 2mentioning

confidence: 99%

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Commutativity in the lattice of topologizing filters of a commutative semiartinian ring

Arega

Berg

2019

Communications in Algebra

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The set FilR R of all right topologizing filters on a fixed but arbitrary ring R admits a monoid operation ':' that is in general noncommutative, even in cases where the ring R is commutative. Earlier results show (see [8]) that commutativity of the monoid operation ':', when imposed as a condition on FilR R, manifests as a type of finiteness condition on R. In a quite separate and much earlier study, Shores [6] has shown that if R is a commutative semiartinian ring, then R will be artinian precisely if the first two terms in the Loewy series for R R , namely soc(R R ) and soc 2 (R R ), are finitely generated. Shores goes further to produce examples which show that the finiteness of just soc(R R ) exercises no constraint whatsoever on the length of R R . The main result of this paper asserts that a commutative semiartinian ring R will be artinian precisely if soc(R R ) is finitely generated and the monoid operation ':' on FilR R is commutative. A family of commutative semiartinian rings of Loewy length 3 is constructed and this used to delineate earlier theory. In particular, and within this family, rings R are exhibited such that (1) soc(R R ) and soc 2 (R R )/soc(R R ) have infinite length, yet (2) the monoid operation ':' on FilR R is commutative

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Section: 2mentioning

confidence: 99%

“…The study of rings R for which Fil R R is commutative was initiated in [11], but later widened in [1] to include rings enjoying the more general two-sided residuation property. Further contributions to this project are made in Section 6.…”

Section: Introductionmentioning

confidence: 99%

Commutativity in the lattice of topologizing filters of a commutative semiartinian ring

Arega

Berg

2019

Communications in Algebra

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Commutativity in the lattice of topologizing filters of a ring – localization and congruences

Arega

Berg

2019

Acta Math. Hungar.

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The order dual [Fil R R ] du of the set Fil R R of all right topologizing filters on a fixed but arbitrary ring R is a complete lattice ordered monoid with respect to the (order dual) of inclusion and a monoid operation ':' that is, in general, noncommutative. It is known that [Fil R R ] du is always left residuated, meaning, for each pair F, G ∈ Fil R R there exists a smallest H ∈ Fil R R such that H : G ⊇ F, but is not, in general, right residuated (there exists a smallest H such that G : H ⊇ F). Rings R for which [Fil R R ] du is both left and right residuated are shown to satisfy the DCC on left annihilator ideals and possess only finitely many minimal prime ideals.It is shown that every maximal ideal P of a commutative ring R gives rise to an onto homomorphism of lattice ordered monoids φP from [Fil R] du to [Fil R P ] du where R P denotes the localization of R at P . The kernel ≡ φP of φP is a congruence on [Fil R] du whose properties we explore. Defining Rad(Fil R) to be the intersection of all congruences ≡ φP as P ranges through all maximal ideals of R, we show that for commutative VNR rings R, Rad(Fil R) is trivial (the identity congruence) precisely if R is noetherian (and thus a finite product of fields). It is shown further that for arbitrary commutative rings R, Rad(Fil R) is trivial whenever Fil R is commutative (meaning, the monoid operation ':' on Fil R is commutative). This yields, for such rings R, a subdirect embedding of [Fil R] du into the product of all [Fil R P ] du as P ranges through all maximal ideals of R. The theory developed is used to prove that a Prüfer domain R for which Fil R is commutative, is necessarily Dedekind.

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Two-sided residuation in the set of topologizing filters on a ring

Cited by 2 publications

References 10 publications

Commutativity in the lattice of topologizing filters of a commutative semiartinian ring

Commutativity in the lattice of topologizing filters of a commutative semiartinian ring

Commutativity in the lattice of topologizing filters of a ring – localization and congruences

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