Semigroup Rings Research Articles

The tiny trace ideals of the canonical modules in Cohen-Macaulay rings of dimension one

We study one-dimensional Cohen-Macaulay rings whose trace ideal of the canonical module is as small as possible. In this paper we call such rings far-flung Gorenstein rings. We investigate far-flung Gorenstein rings in relation with the endomorphism algebras of the maximal ideals and numerical semigroup rings. We show that the solution of the Rohrbach problem in additive number theory provides an upper bound for the multiplicity of far-flung Gorenstein numerical semigroup rings. Reflexive modules over far-flung Gorenstein rings are also studied.

Hausdorff series in semigroup rings of rectangular bands

"The Hausdorff series provides a solution to the equation $w=\log(e^ue^v)$ given by a recursive formula which can be expressed as nested commutators of $u$ and $v$. Evolutions of the Haussdorff series in various algebras and rings has been considered in obtaining a closed form of this formula. We consider the rectangular band $L_m\times R_n$ determined by the left zero semigroup $L_m$ and the right zero semigroup $R_n$ of order $m$ and $n$, respectively. Let $\mathbb R\langle L_m\times R_n\rangle$ be the semigroup ring spanned on $L_m\times R_n$ together with the identity element $1$. We provide a closed form of the formula for solving the equation in $\mathbb R\langle L_m\times R_n\rangle$."

On the algebraic invariants of certain affine semigroup rings

On the algebraic invariants of certain affine semigroup rings

Integral closure, basically full closure, and duals of nonresidual closure operations

We develop a duality for operations on nested pairs of modules that generalizes the duality between absolute interior operations and residual closure operations from [5], extending our previous results to the expanded context. We apply this duality in particular to integral and basically full closures and their respective cores to obtain integral and basically empty interiors and their respective hulls. We also dualize some of the known formulas for the core of an ideal to obtain formulas for the hull of a submodule of the injective hull of the residue field. The article concludes with illustrative examples in a numerical semigroup ring.

Standard pairs for monomial ideals in semigroup rings

We extend the notion of standard pairs to the context of monomial ideals in semigroup rings. Standard pairs can be used as a data structure to encode such monomial ideals, providing an alternative to generating sets that is well suited to computing intersections, decompositions, and multiplicities. We give algorithms to compute standard pairs from generating sets and vice versa and make all of our results effective. We assume that the underlying semigroup ring is positively graded, but not necessarily normal. The lack of normality is at the root of most challenges, subtleties, and innovations in this work.

Affine semigroups of maximal projective dimension

A submonoid of \( {\mathbb {N}}^d \) is of maximal projective dimension (\({\text {MPD}}\)) if the associated affine semigroup ring has the maximum possible projective dimension. Such submonoids have a nontrivial set of pseudo-Frobenius elements. We generalize the notion of symmetric semigroups, pseudo-symmetric semigroups, and row-factorization matrices for pseudo-Frobenius elements of numerical semigroups to the case of \({\text {MPD}}\)-semigroups in \({\mathbb {N}}^d\). Under suitable conditions, we prove that these semigroups satisfy the generalized Wilf’s conjecture. We prove that the generic nature of the defining ideal of the associated semigroup ring of an \({\text {MPD}}\)-semigroup implies uniqueness of row-factorization matrix for each pseudo-Frobenius element. Further, we give a description of pseudo-Frobenius elements and row-factorization matrices of gluing of \({\text {MPD}}\)-semigroups. We prove that the defining ideal of gluing of \({\text {MPD}}\)-semigroups is never generic.

Ulrich ideals in numerical semigroup rings of small multiplicity

Ulrich ideals in numerical semigroup rings of small multiplicity are studied. If the semigroups are three-generated but not symmetric, the semigroup rings are Golod, since the Betti numbers of the residue class fields of the semigroup rings form an arithmetic progression; therefore, these semigroup rings are G-regular ([15]), possessing no Ulrich ideals. Nevertheless, even in the three-generated case, the situation is different, when the semigroups are symmetric. We shall explore this phenomenon, describing an explicit system of generators, that is the normal form of generators, for the Ulrich ideals in the numerical semigroup rings of multiplicity at most 3. As the multiplicity is greater than 3, in general the task of determining all the Ulrich ideals seems formidable, which we shall experience, analyzing one of the simplest examples of semigroup rings of multiplicity 4.

Semigroup rings as weakly Krull domains

Let $D$ be an integral domain and $\Gamma$ be a torsion-free commutative cancellative (additive) semigroup with identity element and quotient group $G$. In this paper, we show that if char$(D)=0$ (resp., char$(D)=p>0$), then $D[\Gamma]$ is a weakly Krull domain if and only if $D$ is a weakly Krull UMT-domain, $\Gamma$ is a weakly Krull UMT-monoid, and $G$ is of type $(0,0,0, \dots )$ (resp., type $(0,0,0, \dots )$ except $p$). Moreover, we give arithmetical applications of this result.

On the Gorenstein locus of simplicial affine semigroup rings

We study the Gorenstein locus of simplicial affine semigroup rings in terms of some Apéry sets. The results come from an analysis of Cohen-Macaulay type of homogeneous localizations at monomial prime ideals. Characterizing when the homogeneous localization at a monomial prime ideal is Gorenstein, we can read the dimension of the non-Gorenstein locus. In particular, we determine when the semigroup ring is Gorenstein on the punctured spectrum.

Simplicial Affine Semigroups with Monomial Minimal Reduction Ideals

We characterize when the monomial maximal ideal of a simplicial affine semigroup ring has a monomial minimal reduction. When this is the case, we study the Cohen–Macaulay and Gorenstein properties of the associated graded ring and provide several bounds for the reduction number with respect to the monomial minimal reduction.

Type and conductor of simplicial affine semigroups

We provide a generalization of pseudo-Frobenius numbers of numerical semigroups to the context of the simplicial affine semigroups. In this way, we characterize the Cohen-Macaulay type of the simplicial affine semigroup ring K [ S ] . We define the type of S , type ( S ) , in terms of some Apéry sets of S and show that it coincides with the Cohen-Macaulay type of the semigroup ring, when K [ S ] is Cohen-Macaulay. If K [ S ] is a d -dimensional Cohen-Macaulay ring of embedding dimension at most d + 2 , then type ( S ) ≤ 2 . Otherwise, type ( S ) might be arbitrary large and it has no upper bound in terms of the embedding dimension. Finally, we present a generating set for the conductor of S as an ideal of its normalization.

The Schreier property and the composite semigroup ring A + B[Γ*

Let [Formula: see text] be an extension of integral domains and [Formula: see text] a commutative, additive, cancellative torsion-free monoid. Let [Formula: see text] be the semigroup ring of [Formula: see text] over [Formula: see text] and set [Formula: see text]. Suppose that [Formula: see text]. Then [Formula: see text] is a subring of [Formula: see text]. In this paper, we study primal elements in [Formula: see text] domains. As an application, we characterize when the construction [Formula: see text] is a (pre-)Schreier domain.

Standard pairs of monomial ideals over nonnormal affine semigroups in SageMath

We present stdPairs.spyx, a SageMath library to compute standard pairs of a monomial ideal over a pointed (non-normal) affine semigroup ring. Moreover, stdPairs.spyx provides the associated prime ideals, the corresponding multiplicities, and an irredundant irreducible primary decomposition of a monomial ideal. The library expands on the standardPairs function on Macaulay2 over polynomial rings, and is based on algorithms from arXiv:2005.10968v2 [math.AC]. We also provide methods that allow the outputs from this library to be compatible with the Normaliz package of Macaulay2 and SageMath.

TWISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY

AbstractTwisted étale groupoid algebras have recently been studied in the algebraic setting by several authors in connection with an abstract theory of Cartan pairs of rings. In this paper we show that extensions of ample groupoids correspond in a precise manner to extensions of Boolean inverse semigroups. In particular, discrete twists over ample groupoids correspond to certain abelian extensions of Boolean inverse semigroups, and we show that they are classified by Lausch’s second cohomology group of an inverse semigroup. The cohomology group structure corresponds to the Baer sum operation on twists.We also define a novel notion of inverse semigroup crossed product, generalizing skew inverse semigroup rings, and prove that twisted Steinberg algebras of Hausdorff ample groupoids are instances of inverse semigroup crossed products. The cocycle defining the crossed product is the same cocycle that classifies the twist in Lausch cohomology.

Coefficient rings of numerical semigroup algebras

Numerical semigroup rings are investigated from the relative viewpoint. It is known that algebraic properties such as singularities of a numerical semigroup ring are properties of a flat numerical semigroup algebra. In this paper, we show that arithmetic and set-theoretic properties of a numerical semigroup ring are properties of an equi-gcd numerical semigroup algebra.

On graded pseudo-valuation domains

Let Γ be a torsionless commutative cancellative monoid and be a Γ-graded integral domain. In this paper, we provide necessary and sufficient conditions for R to be a graded pseudo-valuation domain. In particular, we determine when the semigroup ring over an integral domain D is a graded pseudo-valuation domain. Also some connections of graded pseudo-valuation domains and UMT-domains are studied.

Idempotents and Ideals of Regular Rings

Multiplicative semigroups of rings form an important class of semigroups and one theme in the study of semigroups is how the structure of this semigroup affects the structure of the ring. An important tool in analyzing the structure of a semigroup is the Green’s relations. We study some properties of these relations on the multiplicative semigroup of a regular ring with unity. This also gives easier proofs of some known results on ring theory.

Canonical trace ideal and residue for numerical semigroup rings

For a numerical semigroup ring K[H] we study the trace of its canonical ideal. The colength of this ideal is called the residue of H. This invariant measures how far is H from being symmetric, i.e. how far is K[H] from being a Gorenstein ring. We remark that the canonical trace ideal contains the conductor ideal, and we study bounds for the residue. For 3-generated numerical semigroups we give explicit formulas for the canonical trace ideal and the residue of H. Thus, in this setting we can classify those whose residue is at most one (the nearly Gorenstein ones), and we show the eventual periodic behaviour of the residue in a shifted family.

The generators, relations and type of the Backelin semigroup

We present an explicit minimal set of generators for the defining ideal of the family of Backelin semigroups and find its Betti numbers. In particular, we compute the type of the semigroup as well, correcting a claim in the literature. Additionally, we show that the Betti numbers of the corresponding numerical semigroup ring coincide to those of its tangent cone.

A constructive approach to one-dimensional Gorenstein ${\mathbf {k}}$-algebras

Let $R$ be the power series ring or the polynomial ring over a field $k$ and let $I $ be an ideal of $R.$ Macaulay proved that the Artinian Gorenstein $k$-algebras $R/I$ are in one-to-one correspondence with the cyclic $R$-submodules of the divided power series ring $\Gamma. $ The result is effective in the sense that any polynomial of degree $s$ produces an Artinian Gorenstein $k$-algebra of socle degree $s.$ In a recent paper, the authors extended Macaulay's correspondence characterizing the $R$-submodules of $\Gamma $ in one-to-one correspondence with Gorenstein d-dimensional $k$-algebras. However, these submodules in positive dimension are not finitely generated. Our goal is to give constructive and finite procedures for the construction of Gorenstein $k$-algebras of dimension one and any codimension. This has been achieved through a deep analysis of the $G$-admissible submodules of $\Gamma. $ Applications to the Gorenstein linkage of zero-dimensional schemes and to Gorenstein affine semigroup rings are discussed.