Semigroup Rings Research Articles

The structure of the minimal free resolution of semigroup rings obtained by gluing

We construct a minimal free resolution of the semigroup ring k[C] in terms of minimal resolutions of k[A] and k[B] when 〈C〉 is a numerical semigroup obtained by gluing two numerical semigroups 〈A〉 and 〈B〉. Using our explicit construction, we compute the Betti numbers, graded Betti numbers, regularity and Hilbert series of k[C], and prove that the minimal free resolution of k[C] has a differential graded algebra structure provided the resolutions of k[A] and k[B] possess them. We discuss the consequences of our results in small embedding dimensions. Finally, we give an extension of our main result to semigroups in Nn.

Pseudo-Frobenius numbers versus defining ideals in numerical semigroup rings

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.

Graded Steinberg algebras and partial actions

Given a graded ample Hausdorff groupoid, we realise its graded Steinberg algebra as a partial skew inverse semigroup ring. We use this to show that for a partial action of a discrete group on a locally compact Hausdorff topological space which is totally disconnected, the Steinberg algebra of the associated groupoid is graded isomorphic to the corresponding partial skew group ring. We show that there is a one-to-one correspondence between the open invariant subsets of the topological space and the graded ideals of the partial skew group ring. We also consider the algebraic version of the partial C⁎-algebra of an abelian group and realise it as a partial skew group ring via a partial action of the group on a topological space. Applications to the theory of Leavitt path algebras are given.

Dilatations of numerical semigroups

This paper is focused on numerical semigroups and presents a simple construction, that we call “dilatation”, which, from a starting semigroup S, permits to get an infinite family of semigroups which share several properties with S. The invariants of each semigroup T of this family are given in terms of the corresponding invariants of S and both the Apery set and the minimal generators of T are described. We also study three properties that are close to the Gorenstein property of the associated semigroup ring: almost Gorenstein, 2-AGL, and nearly Gorenstein properties. More precisely, we prove that S satisfies one of these properties if and only if each dilatation of S satisfies the corresponding one.

Local Okounkov bodies and limits in prime characteristic

This article is concerned with the asymptotic behavior of certain sequences of ideals in rings of prime characteristic. These sequences, which we call p-families of ideals, are ubiquitous in prime characteristic commutative algebra (e.g., they occur naturally in the theories of tight closure, Hilbert–Kunz multiplicity, and F-signature). We associate to each p-family of ideals an object in Euclidean space that is analogous to the Newton–Okounkov body of a graded family of ideals, which we call a p-body. Generalizing the methods used to establish volume formulas for the Hilbert–Kunz multiplicity and F-signature of semigroup rings, we relate the volume of a p-body to a certain asymptotic invariant determined by the corresponding p-family of ideals. We apply these methods to obtain new existence results for limits in positive characteristic, an analogue of the Brunn–Minkowski theorem for Hilbert–Kunz multiplicity, and a uniformity result concerning the positivity of a p-family.

Recent results on weakly factorial domains

In this paper, we will survey recent results on weakly factorial domains base on the results of [11, 13, 14]. LetD be an integral domain, X be an indeterminate over D, d ∈ D, R = D[X,d/X] be a subring of the Laurent polynomial ring D[X,1/X], Γ be a nonzero torsionless commutative cancellative monoid with quotient group G, and D[Γ] be the semigroup ring of Γ over D. Among other things, we show that R is a weakly factorial domain if and only if D is a weakly factorial GCD‐domain and d = 0, d is a unit of D or d is a prime element of D. We also show that if char(D) = 0 (resp., char(D) = p > 0), then D[Γ] is a weakly factorial domain if and only if D is a weakly factorial GCD domain, Γ is a weakly factorial GCD semigroup, and G is of type (0,0,0,…) (resp., (0,0,0,…) except p).

Bernstein–Sato Polynomials on Normal Toric Varieties

We generalize the Bernstein–Sato polynomials of Budur, Mustaţǎ, and Saito to ideals in normal semigroup rings. In the case of monomial ideals, we also relate the roots of the Bernstein–Sato polynomial to the jumping coefficients of the corresponding multiplier ideals. To prove the latter result, we obtain a new combinatorial description for the multiplier ideals of a monomial ideal in a normal semigroup ring.

The interplay between Steinberg algebras and skew rings

We study the interplay between Steinberg algebras and skew rings: For a partial action of a group in a Hausdorff, locally compact, totally disconnected topological space, we realize the associated partial skew group ring as a Steinberg algebra (over the transformation groupoid attached to the partial action). We then apply this realization to characterize diagonal preserving isomorphisms of partial skew group rings, over commutative algebras, in terms of continuous orbit equivalence of the associated partial actions. Finally, we show that any Steinberg algebra, associated to a Hausdorff ample groupoid, can be seen as a partial skew inverse semigroup ring.

Semigroup rings as almost Prüfer v-multiplication domains

Let D be an integral domain with quotient field K, \(\Gamma \) a nonzero torsion-free grading monoid and \(\Gamma ^*=\Gamma {\setminus } \{0\}\). In this paper, we characterize when the semigroup ring \(D[\Gamma ]\) is an almost Prufer v-multiplication domain or an almost Prufer domain. We also give an equivalent condition for the composite semigroup ring \(D+K[\Gamma ^*]\) to be an almost Prufer v-multiplication domain or an almost Prufer domain when \(\Gamma \cap -\Gamma =\{0\}\).

Atomic and AP semigroup rings $F[X;M]$, where $M$ is a submonoid of the additive monoid of nonnegative rational numbers

We investigate the atomicity and the AP property of the semigroup rings $F[X;M]$, where $F$ is a field, $X$ is a variable and $M$ is a submonoid of the additive monoid of nonnegative rational numbers. The main notion that we introduce for the purpose of the investigation is the notion of essential generators of $M$.

On semigroup rings with decreasing Hilbert function

Given a one-dimensional semigroup ring R=k[[S]], in this article we study the behaviour of the Hilbert function HR. By means of the notion of support of the elements in S, for some classes of semigroup rings we give conditions on the generators of S in order to have decreasing HR. When the embedding dimension v and the multiplicity e verify v+3≤e≤v+4, the decrease of HR gives an explicit description of the Apéry set of S. In particular for e=v+3, we prove that HR is non-decreasing if e≤12 and we classify the semigroups with e=13 and HR decreasing. Finally we deduce that HR is non-decreasing for every Gorenstein semigroup ring with e≤v+4. This fact is not true in general: through numerical duplication and some of the above results another recent paper shows the existence of infinitely many one-dimensional Gorenstein rings with decreasing Hilbert function.

Doset Hibi rings with an application to invariant theory

ABSTRACTWe define the concept of a doset Hibi ring and a generalized doset Hibi ring which are subrings of a Hibi ring and are normal affine semigroup rings. We apply the theory of (generalized) doset Hibi rings to analyze the rings of absolute orthogonal invariants and absolute special orthogonal invariants and show that these rings are normal and Cohen-Macaulay and has rational singularities if the characteristic of the base field is zero and is F-rational otherwise. We also state criteria of Gorenstein property of these rings.

Noether resolutions in dimension 2

Let R:=K[x1,…,xn] be a polynomial ring over an infinite field K, and let I⊂R be a homogeneous ideal with respect to a weight vector ω=(ω1,…,ωn)∈(Z+)n such that dim(R/I)=d. In this paper we study the minimal graded free resolution of R/I as A-module, that we call the Noether resolution of R/I, whenever A:=K[xn−d+1,…,xn] is a Noether normalization of R/I. When d=2 and I is saturated, we give an algorithm for obtaining this resolution that involves the computation of a minimal Gröbner basis of I with respect to the weighted degree reverse lexicographic order. In the particular case when R/I is a 2-dimensional semigroup ring, we also describe the multigraded version of this resolution in terms of the underlying semigroup. Whenever we have the Noether resolution of R/I or its multigraded version, we obtain formulas for the corresponding Hilbert series of R/I, and when I is homogeneous, we obtain a formula for the Castelnuovo–Mumford regularity of R/I. Moreover, in the more general setting that R/I is a simplicial semigroup ring of any dimension, we provide its Macaulayfication.As an application of the results for 2-dimensional semigroup rings, we provide a new upper bound for the Castelnuovo–Mumford regularity of the coordinate ring of a projective monomial curve. Finally, we describe the multigraded Noether resolution and the Macaulayfication of either the coordinate ring of a projective monomial curve C⊆PKn associated to an arithmetic sequence or the coordinate ring of any canonical projection πr(C) of C to PKn−1.

Affine semigroup rings are of finite F-representation type

ABSTRACTIt is known that normal affine semigroup rings are of finite F-representation type. In this paper, we prove that non-normal affine semigroup rings are also of finite F-representation type.

Graded-valuation domains

ABSTRACTLet Γ be a torsionless grading monoid, a Γ-graded integral domain, H the set of nonzero homogeneous elements of R, K the quotient field of R0, and G0 = Γ∩−Γ the group of units of Γ. We say that R is a graded-valuation domain if either x∈R or x−1∈R for every nonzero homogeneous element x∈RH. In this paper, we show that R is a graded-valuation domain if and only ifΓ is a valuation monoid,Rα = Kx for every 0≠x∈Rα whenever α is not a unit of Γ, and is a graded-valuation domain.Let R = Kγ[X;Γ] be a twisted semigroup ring of Γ over K, C a totally ordered (additive) abelian group, C′ a subgroup of C, μ:K→C′∪{∞} a valuation, φ:Γ→C a function such that C′∪φ(Γ) generates C, and v:R→C∪{∞} the function defined by for every . We show that v is a valuation if and only if μ(γ(a,b))+φ(a+b) = φ(a)+φ(b) for every a,b∈Γ.

FP-injective semirings, semigroup rings and Leavitt path algebras

ABSTRACTIn this paper we give characterisations of FP-injective semirings (previously termed “exact” semirings in work of the first author). We provide a basic connection between FP-injective semirings and P-injective semirings, and establish that FP-injectivity of semirings is a Morita invariant property. We show that the analogue of the Faith-Menal conjecture (relating FP-injectivity and self-injectivity for rings satisfying certain chain conditions) does not hold for semirings. We prove that the semigroup ring of a locally finite inverse monoid over an FP-injective ring is FP-injective and give a criterion for the Leavitt path algebra of a finite graph to be FP-injective.

Semigroup rings and group rings with large center

A ring R is called a ring with a large center or an IIC-ring if any nonzero ideal of R has a nonzero intersection with the center of R. We consider conditions which guarantee that a semigroup ring over an IIC-ring is an IIC-ring.

Characterization of cyclic codes over {ℬ[X;(1/m)Z0]}m > 1 and efficient encoding decoding algorithm for cyclic codes

ABSTRACTThe aim of the paper is twofold. Firstly, cyclic codes of arbitrary length n over the family of semigroup rings are completely characterized in terms of ideals of the rings , where is finite unitary commutative ring. Then, generator matrix for cyclic codes over is derived from their ideal representation. Secondly, an efficient encoding decoding algorithm is presented for polynomial cyclic code of length n based on cyclic code over . Rigorous analyses are performed to examine the efficiency of proposed scheme. Analyses reveal that the proposed algorithm can simultaneously encode and decode m messages of . Furthermore, the proposed technique is capable of detecting and correcting all burst errors up to length md−m and from transmitted codewords of . The newly developed algorithm is also compared with some of the existing coding techniques. It is evident from comparison that the proposed scheme is better than existing coding techniques in Babu and Zimmermann [Decoding of linear codes over galois rings, IEEE Trans. Inform. Theory, 47 (2001)], Nagpaul and Jain [Topics in Applied Algebra, The Brooks/Cole Series in Advanced Mathematics, 2005], Shah et al. [A method for improving the code rate and error correction capability of a cyclic code, Comput. Appl. Math. 32 (2013), pp. 261–274, doi:10.1007/s40314-013-0010-1] and Shah et al. [A decoding method of an n length binary BCH code through (n+1)n length binary cyclic code, An. Acad. Brasil. Ciênc. 85 (2013), pp. 863–872].

Torsion and tensor products over domains and specializations to semigroup rings

Let R be a commutative Noetherian domain, and let M and N be finitely generated R-modules. We give new criteria for determining when the tensor product of two modules has torsion. We also give constructive formulas for the torsion submodule of M⊗RN. In certain cases we determine bounds on the length and minimal number of generators of the torsion submodule. Lastly, we focus on the case where R is a numerical semigroup ring with the goal of making progress on the Huneke–Wiegand Conjecture.

Generalized Krull Semigroup Rings

In this article, we give a complete characterization of when the (composite) semigroup ring is a generalized Krull domain.