Semigroup Rings Research Articles

Almost Gorenstein simplicial semigroup rings

We give a criterion for almost Gorenstein property for semigroup rings associated with simplicial semigroups. We extend Nari's theorem for almost symmetric numerical semigroups to simplicial semigroups with higher rank. By this criterion, we determine 2-dimensional normal semigroup rings which have “Ulrich elements” defined in [8].

The growth recurrence and Gelfand-Kirillov base of the ordinary cusp

We introduce the Gelfand-Kirillov base for a numerical semigroup ring over the prime field of characteristic p p , where p p is prime, and show its existence for the semigroup ring of the ordinary cusp by establishing a growth recurrence with respect to Frobenius.

Class group and factorization in orders of a PID

In this paper, we study properties of factorization in orders of a PID via the computation of algebraic invariants that measure the failure of unique factorization. The focus is on the numerical semigroup rings over a finite field and the orders of imaginary quadratic fields with class number 1. We also give a complete description of the class group structure of those rings.

Families of numerical semigroups and a special case of the Huneke-Wiegand conjecture

The Huneke-Wiegand conjecture is a decades-long open question in commutative algebra. García-Sánchez and Leamer showed that a special case of this conjecture concerning numerical semigroup rings k [ Γ ] can be answered in the affirmative by locating certain arithmetic sequences within the numerical semigroup Γ. In this paper, we use their approach to prove the Huneke-Wiegand conjecture in the case where Γ is generated by a generalized arithmetic sequence and showcase how visualizations can be leveraged to find the requisite arithmetic sequences.

Generalized Loewy length of Cohen-Macaulay local and graded rings

We generalize a theorem of Ding relating the generalized Loewy length gℓℓ(R) and index of a one-dimensional Cohen-Macaulay local ring (R,m,k). Ding proved that if R is Gorenstein, the associated graded ring is Cohen-Macaulay, and k is infinite, then the generalized Loewy length and index of R are equal. However, if k is finite, equality may not hold. We prove that if the index of a one-dimensional Cohen-Macaulay local ring is finite and the associated graded ring has a homogeneous nonzerodivisor of degree t, then gℓℓ(R)≤index(R)+t−1. Next we prove that if R is a one-dimensional hypersurface ring with a witness to the generalized Loewy length that induces a regular initial form on the associated graded ring, then the generalized Loewy length achieves this upper bound. We then compute the generalized Loewy lengths of several families of examples of one-dimensional hypersurface rings over finite fields. Finally, we study a graded version of the generalized Loewy length and determine its value for numerical semigroup rings.

Semigroup rings which are strongly nil-clean

Strongly nil-cleanness and uniquely nil-cleanness are two important clean properties of rings. The aim of this paper is to answer when a semigroup ring is strongly or uniquely nil-clean. We obtain conditions on a coefficient ring together with conditions on a semigroup that completely characterize when the corresponding semigroup ring is strongly nil-clean. Moreover, when the coefficient ring is an integral domain, we characterize when the corresponding semigroup ring is uniquely nil-clean.

Regular t-ideals of polynomial rings and semigroup rings with zero divisors

For an integrally closed domain D with quotient field K, it is well known that every fractional t-ideal of D[X] has the form hI[X] with h∈K(X) and I a t-ideal of D. We generalize this useful result to certain polynomial rings and semigroup rings with zero divisors. As an application of our work, we characterize the semigroup rings with torsion-free cancellative monoids of exponents that are Prüfer v-multiplication rings or (generalized) greatest common divisor rings. By showing that Glaz's (generalized) greatest common divisor ring property ascends to polynomial rings, we affirmatively settle one of her conjectures.

COUNTING GEOMETRIC BRANCHES VIA THE FROBENIUS MAP AND F-NILPOTENT SINGULARITIES

AbstractWe give an explicit formula to count the number of geometric branches of a curve in positive characteristic using the theory of tight closure. This formula readily shows that the property of having a single geometric branch characterizes F-nilpotent curves. Further, we show that a reduced, local F-nilpotent ring has a single geometric branch; in particular, it is a domain. Finally, we study inequalities of Frobenius test exponents along purely inseparable ring extensions with applications to F-nilpotent affine semigroup rings.

Remarks on almost Gorenstein rings

This paper investigates the relation between the almost Gorenstein properties for graded rings and for local rings. Once R is an almost Gorenstein graded ring, the localization RM of R at the graded maximal ideal M is almost Gorenstein as a local ring. The converse does not hold true in general ([7, Theorems 2.7, 2.8], [8, Example 8.8]). However, it does for one-dimensional graded domains with mild conditions, which we clarify in the present paper. We explore the defining ideals of almost Gorenstein numerical semigroup rings as well.

Inverse polynomials of numerical semigroup rings

We study the defining ideal of a numerical semigroup ring k[H] using the inverse polynomial attached to the Artinian ring k[H]/(th). We give a criterion for H to be symmetric or almost symmetric using the annihilator of the inverse system. Also we give characterizations of symmetric numerical semigroups with small multiplicity and give a new proof of Bresinsky's Theorem for symmetric semigroups generated by 4 elements.

On the depth of simplicial affine semigroup rings

AbstractWe recall and delve into the different characterizations of the depth of an affine semigroup ring, providing an original characterization of depth two in three and four dimensional cases which are closely related to the existence of a maximal element in certain Apéry sets.

The Koszulness of numerical semigroup rings of minimal multiplicity

Let R = k [H] be a numerical semigroup ring over a field k and grm (R) is the associated graded ring of R. In this paper, we show that grm (R) is a Cohen-Macaulay ring, provided H has minimal multiplicity. As a consequence, we conclude that the numerical semigroup ring R = k [H] of minimal multiplicity is a Koszul ring, i.e., the residue field k has a grm (R) – linear free resolution.

Graded local cohomology of modules over semigroup rings

We give a combinatorial description of local cohomology modules of a graded module over a semigroup ring, with support at the graded maximal ideal. This combinatorial framework yields Hochster-type formulas for the Hilbert series of such local cohomology modules in terms of the homology of finitely many polyhedral cell complexes. A Cohen–Macaulay criterion immediately follows. We also provide an alternative proof of a result of [18] characterizing Cohen–Macaulay affine semigroup rings.

Levelness versus nearly Gorensteinness of homogeneous rings

Levelness and nearly Gorensteinness are well-studied properties of graded rings as a generalized notion of Gorensteinness. In this paper, we compare the strength of these properties. For any Cohen-Macaulay homogeneous affine semigroup ring R, we give a necessary condition for R to be non-Gorenstein and nearly Gorenstein in terms of the h-vector of R and we show that if R is nearly Gorenstein with Cohen-Macaulay type 2, then it is level. We also show that if Cohen-Macaulay type is more than 2, there are 2-dimensional counterexamples. Moreover, we characterize nearly Gorensteinness of Stanley-Reisner rings of low-dimensional simplicial complexes.

Purity of monoids and characteristic-free splittings in semigroup rings

Inspired by methods in prime characteristic in commutative algebra, we introduce and study combinatorial invariants of seminormal monoids. We relate such numbers with the singularities and homological invariants of the semigroup ring associated to the monoid. Our results are characteristic independent.

Differentially fixed ideals in affine semigroup rings

We give a complete characterization of when monomial ideals are fixed by differential operators of affine semigroup rings over [Formula: see text]. Perhaps surprisingly, every monomial ideal is fixed by an infinite set of homogeneous differential operators and is in fact determined by them. This opens up a new tool for studying monomial ideals. We explore applications of this to (mixed) multiplier ideals and other variants as well as give examples of detecting ideal membership in integrally closed powers and symbolic powers of squarefree monomial ideals.

On the associated graded ring of semigroup algebras

In this paper, we give a necessary and sufficient condition for the Cohen-Macaulayness of the associated graded ring of a simplicial affine semigroup using Gröbner basis. We generalize the concept of homogeneous numerical semigroup for the simplicial affine semigroup and show that the Betti numbers of the corresponding semigroup ring matches with the Betti numbers of the associated graded ring. We also discuss nice extensions for simplicial affine semigroups, a concept which is motivated by nice extensions of numerical semigroups.

Atomic semigroup rings and the ascending chain condition on principal ideals

An integral domain is called atomic if every nonzero nonunit element factors into irreducibles. On the other hand, an integral domain is said to satisfy the ascending chain condition on principal ideals (ACCP) if every ascending chain of principal ideals stabilizes. It was asserted by P. Cohn back in the sixties that every atomic domain satisfies the ACCP, but such an assertion was refuted by A. Grams in the seventies with a neat counterexample. Still, atomic domains without the ACCP are notoriously elusive, and just a few classes have been found since Grams’ first construction. In the first part of this paper, we generalize Grams’ construction to provide new classes of atomic domains without the ACCP. In the second part of this paper, we construct a new class of atomic semigroup rings without the ACCP.

Almost symmetric good semigroups

The class of good semigroups is a class of subsemigroups of Nh\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {N}}^h$$\\end{document}, that includes the value semigroups of rings associated to curve singularities and their blowups, and allows to study combinatorically the properties of these rings. In this paper we give a characterization of almost symmetric good subsemigroups of Nh\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {N}}^h$$\\end{document}, extending known results in numerical semigroup theory and in one-dimensional ring theory, and we apply these results to obtain new results on almost Gorenstein one-dimensional analytically unramified rings.

Publisher Correction: On the algebraic invariants of certain affine semigroup rings

Publisher Correction: On the algebraic invariants of certain affine semigroup rings