When multiplication of topologizing filters is commutative

Berg, John E. van den

doi:10.1016/s0022-4049(99)00033-x

Cited by 8 publications

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“…In Section 7, theory is put to use to show that the only Prüfer domains R for which Fil R is commutative are noetherian and thus Dedekind domains. This result generalises [11,Corollary 32,page 102].…”

Section: Introductionsupporting

confidence: 77%

“…With reference to the implication (a) ⇒(b) of the previous proposition, we point out that if the requirement that R is VNR is dispensed with, then Statement (b) holds under conditions much weaker than (a). Indeed, within the class of all commutative rings, it is known (see [11,Corollary 8,page 91]) that Fil R is commutative whenever R is noetherian and, as we shall prove in the next section (Theorem 32), Rad(Fil R) is trivial whenever Fil R is commutative. In general, even for commutative rings R, [Fil R R ] du need not be right residuated.…”

Section: Congruences On Fil Rmentioning

confidence: 92%

“…If R is an arbitrary ring, the maps A → A −1 0 and A → 0A −1 represent a Galois connection between the sets of left annihilator ideals of R, and right annihilator ideals of R. Thus R will satisfy the DCC on left annihilator ideals, precisely if it satisfies the ACC on right annihilator ideals. It is shown in [11,Theorem 19,page 98] that for an arbitrary ring R, if Fil R R is commutative, then R satisfies the ACC on right annihilator ideals. The next result, which is an immediate consequence of Proposition 25, the fact that every left annihilator ideal of R is a hereditary pretorsion submodule of R R , and the equivalence of the DCC on left annihilator ideals and ACC on right annihilator ideals, shows that this theorem remains valid if the requirement that Fil R R is commutative is weakened to [Fil R R ] du is two-sided residuated.…”

Section: Residuation and Commutativity Inmentioning

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%

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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: Introductionsupporting

confidence: 77%

Section: Congruences On Fil Rmentioning

confidence: 92%

Section: Residuation and Commutativity Inmentioning

confidence: 99%

Section: Introductionmentioning

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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“…Notice that the operation ':' defined here is opposite to the multiplication operation introduced in[12],[13] and[11]. Consequently, properties which are prefixed with 'left' in the aforementioned papers, become 'right' in this paper.…”

mentioning

confidence: 98%

Duprime and dusemiprime modules

Berg¹,

Wisbauer

2001

Journal of Pure and Applied Algebra

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A lattice ordered monoid is a structure L; ⊕, 0 L ; ≤ where L; ⊕, 0 L is a monoid, L; ≤ is a lattice and the binary operation ⊕ distributes over finite meets. If M ∈ R-Mod then the set IL M of all hereditary pretorsion classes of σ[M ] is a lattice ordered monoid with binary operation given by α : M β := {N ∈ σ[M ] | there exists A ≤ N such that A ∈ α and N/A ∈ β}, whenever α, β ∈ IL M (the subscript in : M is omitted if σ[M ] = R-Mod). σ[M ] is said to be duprime (resp. dusemiprime) if M ∈ α : M β implies M ∈ α or M ∈ β (resp. M ∈ α : M α implies M ∈ α), for any α, β ∈ IL M. The main results characterize these notions in terms of properties of the subgenerator M. It is shown, for example, that M is duprime (resp. dusemiprime) if M is strongly prime (resp. strongly semiprime). The converse is not true in general, but holds if M is polyform or projective in σ[M ]. The notions duprime and dusemiprime are also investigated in conjunction with finiteness conditions on IL M , such as coatomicity and compactness.

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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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When multiplication of topologizing filters is commutative

Cited by 8 publications

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

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

Duprime and dusemiprime modules

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

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