Pseudo-Frobenius numbers versus defining ideals in numerical semigroup rings

Gotō, Shirō; Kien, Do Van; Matsuoka, Naoyuki; Trương, Hoàng Lê

doi:10.1016/j.jalgebra.2018.04.025

Cited by 10 publications

(6 citation statements)

References 13 publications

(23 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The properties of this class of rings have been studied by several authors in the last two decades (cf. [2], [5], [6], [10], [12], [15], [19] for example); in particular, curious patterns started to appear in works dealing with the Cohen-Macaulay type of almost Gorenstein rings. In the original context of one-dimensional analytically unramified rings, including local rings associated to monomial curves, it is well-known that if the embedding dimension is at most 3 then the Cohen-Macaulay type is either 1 (and thus the ring is Gorenstein) or 2 (cf.…”

Section: Introductionmentioning

confidence: 99%

“…Families of almost Gorenstein rings with arbitrarily large Cohen-Macaulay type t are present in the literature (cf. [9], [10]); however, for these families of rings, the embedding dimension e is at least t 2 (actually, in this context there is no example in the literature of an almost Gorenstein ring satisfying t > 2e). Therefore, contrary to the general context, for almost Gorenstein rings and curves the following question (cf.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the type of an almost Gorenstein monomial curve

Moscariello

2016

Journal of Algebra

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the type of an almost Gorenstein monomial curve

Moscariello

2016

Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…One can construct many examples of almost Gorenstein rings (e.g., [4,8,10,12,13,14,15,16,18,19,20,22,27,28,34]). The significant examples of almost Gorenstein rings are one-dimensional Cohen-Macaulay local rings of finite Cohen-Macaulay representation type and two-dimensional rational singularities.…”

Section: Survey On Almost Gorenstein Ringsmentioning

confidence: 99%

Almost Gorenstein rings arising from fiber products

Endo,

Goto,

Isobe

2019

Preprint

Self Cite

View full text Add to dashboard Cite

The purpose of this paper is, as part of the stratification of Cohen-Macaulay rings, to investigate the question of when the fiber products are almost Gorenstein rings. We show that the fiber product R × T S of Cohen-Macaulay local rings R, S of the same dimension d > 0 over a regular local ring T with dim T = d − 1 is an almost Gorenstein ring if and only if so are R and S. Besides, the other generalizations of Gorenstein properties are also explored.

show abstract

“…Let H ⊆ Z be a numerical semigroup and k be a field. Then the numerical semigroup ring k[[H]] often have the form obtained in Proposition 2.6; see, for examples, [10,17]. In particular, the Auslander-Reiten conjecture holds for all three generated numerical semigroup rings.…”

Section: Powers Of Parameter Ideals and Determinantal Ringsmentioning

confidence: 99%

The Auslander-Reiten conjecture for certain non-Gorenstein Cohen-Macaulay rings

Kumashiro¹

2019

Preprint

View full text Add to dashboard Cite

Let R be a Cohen-Macaulay local ring and Q be a parameter ideal of R. Due to M. Auslander, S. Ding, and Ø. Solberg, the Auslander-Reiten conjecture holds for R if and only if it holds for the residue ring R/Q. In the former part of this paper, we study the Auslander-Reiten conjecture for the ring R/Q ℓ in connection with that for R and prove the equivalence of them for the case where R is Gorenstein and ℓ ≤ dim R.In the latter part, we study the existence of Ulrich ideals and generalize the result of maximal embedding dimension by J. Sally. Due to these two of our results, we finally show that the Auslander-Reiten conjecture holds if there is an Ulrich ideal whose residue ring is a complete intersection. Besides them, we also explore the Auslander-Reiten conjecture for determinantal rings.As is well known, non-zerodivisors preserve the Auslander-Reiten conjecture (Proposition 2.1). Hence, through Theorem 1.2, many Gorenstein rings which satisfy the Auslander-Reiten conjecture are given. Even if a given local ring is not Gorenstein, the conjecture still holds if the ring is Golod or almost Gorenstein ([21, Proposition 1.4.] and 2010 Mathematics Subject Classification. 13C40, 13D07, 13H10. Key words and phrases. Auslander-Reiten conjecture, Cohen-Macaulay ring, Gorenstein ring, complete intersection, power of parameter ideal, determinantal ring, Ulrich ideal.The author was partially supported by JSPS KAKENHI Grant Number JP19J10579.

show abstract

Pseudo-Frobenius numbers versus defining ideals in numerical semigroup rings

Abstract: The structure of the defining ideal of the semigroup ring k[H] of a numerical semigroup H over a field k is described, when the pseudo-Frobenius numbers of H are multiples of a fixed integer.2010 Mathematics Subject Classification. 13A02, 13C05, 13D02, 13H10.

Cited by 10 publications

References 13 publications

On the type of an almost Gorenstein monomial curve

On the type of an almost Gorenstein monomial curve

Almost Gorenstein rings arising from fiber products

The Auslander-Reiten conjecture for certain non-Gorenstein Cohen-Macaulay rings

Contact Info

Product

Resources

About