Open Problems in Commutative Ring Theory

Cahen, Paul-Jean; Fontana, Marco; Frisch, Sophie; Glaz, Sarah

doi:10.1007/978-1-4939-0925-4_20

Cited by 25 publications

(13 citation statements)

References 98 publications

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“…In the present paper, we prove that every strongly primary domain is locally tame and we provide a characterization of global tameness (Theorems 3.8 (b) and 3.9.2). In particular, all one-dimensional local Mori domains turn out to be locally tame and this answers Problem 38 in [7] in the affirmative. Although our present approach is semigroup theoretical over large parts (Theorem 3.8 (a)), it also uses substantially the ring structure, and this is unavoidable since strongly primary Mori monoids need not be locally tame as shown in [22,Proposition 3.7 and Example 3.8] (see Example 3.17.1).…”

Section: Introductionmentioning

confidence: 60%

“…(b) By Theorem 3.7, condition (5) implies that f 6 ¼ ð0Þ: Thus, each of the first five conditions implies that f 6 ¼ ð0Þ: Obviously, each of the other four conditions implies that f 6 ¼ ð0Þ: By item a., all the nine conditions are equivalent. In the next corollary, we answer in the positive Problem 38 in [7]. Proof.…”

Section: On the Arithmetic Of Strongly Primary Monoids And Domainsmentioning

confidence: 91%

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On strongly primary monoids and domains

Geroldinger

Roitman

2020

Communications in Algebra

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A commutative integral domain is primary if and only if it is one-dimensional and local. A domain is strongly primary if and only if it is local and each nonzero principal ideal contains a power of the maximal ideal. Hence, one-dimensional local Mori domains are strongly primary. We prove among other results that if R is a domain such that the conductor ðR :RÞ vanishes, then KðRÞ is finite; that is, there exists a positive integer k such that each nonzero nonunit of R is a product of at most k irreducible elements. Using this result, we obtain that every strongly primary domain is locally tame, and that a domain R is globally tame if and only if KðRÞ ¼ 1: In particular, we answer Problem 38 of the open problem list by Cahen et al. in the affirmative. Many of our results are formulated for monoids.

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Section: Introductionmentioning

confidence: 60%

Section: On the Arithmetic Of Strongly Primary Monoids And Domainsmentioning

confidence: 91%

On strongly primary monoids and domains

Geroldinger

Roitman

2020

Communications in Algebra

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“…(3) ⇒ (4) Let p be a maximal ideal of T ) is a T(R[x])-module, and so is flat by (3). Hence M is a τ q -flat R-module by Theorem 4.3.…”

Section: Modules Over T(r[x])mentioning

confidence: 92%

“…In 2014, Cahen et al [3,Problem 1] posed the following open question: Problem 1b: Let R be a total ring of quotients (i.e. R = T(R)).…”

Section: Modules Over T(r[x])mentioning

confidence: 99%

On $τ_q$-flatness and $τ_q$-coherence

Zhang¹

2021

Preprint

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In this paper, we introduce and study the notions of τ q -flat modules and τ q -coheret rings. First, by investigating the Nagata rings of τ q -torsion theory, we show that the small finitistic dimensions of T(R[x]) are all equal to 0 for any ring R. Then, we introduce the notion of τ q -VN regular rings (i.e. over which all modules are τ q -flat), and show that a ring R is a τ q -VN regular ring if and only if T(R[x]) is a von Neumann regular ring. Finally, we obtain the Chase theorem for τ q -coheret rings: a ring R is τ q -coherent if and only if any direct product of R is τ q -flat if and only if any direct product of flat R-modules is τ q -flat. Some examples are provided to compare with the known conceptions.

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“…We show (Proposition 2.7) that the infimum of a compact family of semistar operations of finite type is of finite type, answering a question ! posed in [4], and that the converse is true when each operation of the family is induced by a localization of A or by a valuation ring (Corollary 4.4 and Proposition 4.5). We also conjecture that the converse is true for any family of overrings.…”

Section: Introductionmentioning

confidence: 99%

Some topological considerations on semistar operations

Finocchiaro¹,

Spirito²

2014

Journal of Algebra

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Abstract. We consider properties and applications of a new topology, called the Zariski topology, on the space SStar(A) of all the semistar operations on an integral domain A. We prove that the set of all overrings of A, endowed with the classical Zariski topology, is homeomorphic to a subspace of SStar(A). The topology on SStar(A) provides a general theory, through which we see several algebraic properties of semistar operation as very particular cases of our construction. Moreover, we show that the subspace SStar f (A) of all the semistar operations of finite type on A is a spectral space.

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Open Problems in Commutative Ring Theory

Cited by 25 publications

References 98 publications

On strongly primary monoids and domains

On strongly primary monoids and domains

On $τ_q$-flatness and $τ_q$-coherence

Some topological considerations on semistar operations

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