Semigroup Rings Research Articles

Note on the divisoriality of domains of the form $k[[X^{p}, X^{q}]]$, $k[X^{p}, X^{q}]$, $k[[X^{p}, X^{q}, X^{r}]]$, and $k[X^{p}, X^{q}, X^{r}

Let $k$ be a field and $X$ an indeterminate over $k$. In this note we prove that the domain $k[[X^{p}, X^{q}]]$ (resp. $k[X^{p}, X^{q}]$) where $p, q$ are relatively prime positive integers is always divisorial but $k[[X^{p}, X^{q}, X^{r}]]$ (resp. $k[X^{p}, X^{q}, X^{r}]$) where $p, q, r$ are positive integers is not. We also prove that $k[[X^{q}, X^{q+1}, X^{q+2}]]$ (resp. $k[X^{q}, X^{q+1}, X^{q+2}]$) is divisorial if and only if $q$ is even. These are very special cases of well-known results on semigroup rings, but our proofs are mainly concerned with the computation of the dual (equivalently the inverse) of the maximal ideal of the ring.

On the Hilbert function of the tangent cone of a monomial curve

In this paper we study the Hilbert function of \(\mathrm {gr}_{\mathfrak {m}}(R)\), when R is a numerical semigroup ring or, equivalently, the coordinate ring of a monomial curve. In particular, we prove a sufficient condition for a numerical semigroup ring in order get a non-decreasing Hilbert function, without making any assumption on its embedding dimension; moreover, we show how this new condition allows us to improve known results about this problem. To this end we use certain invariants of the semigroup, with particular regard to its Apery-set.

A proof of Schwartz's conjecture about the eigenvalues of Lannes' T-functor

This paper gives an algebraic proof of a conjecture due to Lionel Schwartz which asserts that the linear operator induced by Lannes' T-functor on the Grothendieck group Knred generated by indecomposable summands of the mod p cohomology of a rank n elementary abelian p-group is diagonalizable over Q, with eigenvalues 1,p,…,pn and with multiplicities pn−pn−1,pn−1−pn−2,…,p−1,1, respectively. Using work of Harris and Shank, we first reduce this to an algebraic question involving the Grothendieck ring G0(Mn,p) of modules over the semigroup ring Fp[End(Fp⊕n)], showing that the induced action of T on Knred corresponds to the multiplication by an explicit element. In the second step, we establish the separability of the algebra C⊗G0(Mn,p), from which the diagonalizability and the computation of the eigenvalues and their multiplicities follow easily. The arguments use ingredients from the theory of Brauer characters of finite groups.

Semi-Baer modules

In this paper, we introduce the concepts of semi-Baer, semi-quasi Baer, semi-p.q. Baer and semi-p.p. modules as a generalization of Baer, quasi Baer, p.q. Baer and p.p. modules respectively. To define these concepts, we introduce concepts of multiplicative order of an element and a multiplicatively finite element in rings. Further, we characterize these concepts in modules over reduced rings. Also, it is proved that semi-Baer and semi-quasi Baer properties are preserved by polynomial extensions and power series extensions of modules. It is proved that for a ring R and a monoid G, if the semi group ring RG is semi-Baer (semi-quasi Baer) then so is R.

TRACES ON SEMIGROUP RINGS AND LEAVITT PATH ALGEBRAS

AbstractThe trace on matrix rings, along with the augmentation map and Kaplansky trace on group rings, are some of the many examples of linear functions on algebras that vanish on all commutators. We generalize and unify these examples by studying traces on (contracted) semigroup rings over commutative rings. We show that every such ring admits a minimal trace (i.e., one that vanishes only on sums of commutators), classify all minimal traces on these rings, and give applications to various classes of semigroup rings and quotients thereof. We then study traces on Leavitt path algebras (which are quotients of contracted semigroup rings), where we describe all linear traces in terms of central maps on graph inverse semigroups and, under mild assumptions, those Leavitt path algebras that admit faithful traces.

Arithmetic of Semigroup Semirings

We define semigroup semirings by analogy with group rings and semigroup rings. We study the arithmetic properties and determine sufficient conditions under which a semigroup semiring is atomic, has finite factorization, or has bounded factorization. We also present a semigroup-semiring analog (although not a generalization) of the Gauss lemma on primitive polynomials.

Krull Dimension of Monomial Ideals in Polynomial Rings with Real Exponents

We develop a new technique for studying monomial ideals in the standard polynomial rings A[X 1,…, X d ] where A is a commutative ring with identity. The main idea is to consider induced ideals in the semigroup ring where 𝕄1,…, 𝕄 d are nonzero additive subgroups of ℝ. We prove that the set of nonzero finitely generated monomial ideals in R has the structure of a metric space, and we prove that a version of Krull dimension for this setting is lower semicontinuous with respect to this metric space structure. We also show how to use discrete techniques to study certain monomial ideals in this context.

S-Noetherian Properties of Composite Ring Extensions

Let R be a commutative ring with identity and S a multiplicative subset of R. We say that R is an S-Noetherian ring if for each ideal I of R, there exist an s ∈ S and a finitely generated ideal J of R such that sI ⊆ J ⊆ I. In this article, we study transfers of S-Noetherian property to the composite semigroup ring and the composite generalized power series ring.

Neutrosophic LA-Semigroup Rings

Neutrosophic LA-semigroup is a midway structure between a neutrosophic groupoid and a commu- tative neutrosophic semigroup. Rings are the old concept in algebraic structures. We combine the neutrosophic LA-semigroup and ring together to form the notion of neutrosophic LA-semigroup ring. Neutrosophic LA- semigroup ring is defined analogously to neutrosophic group ring and neutrosophic semigroup ring.

Residual Complex on the Tangent Cone of a Numerical Semigroup Ring

A residual complex on the tangent cone of a numerical semigroup ring is constructed in terms of modules of homomorphisms. Macaulay’s inverse system is generalized. Applications of the constructions include a criterion for the Gorensteiness of the tangent cone in terms of the symmetry of the set of basis monomials.

Generic representation theory of finite fields in nondescribing characteristic

Let Rep(F;K) denote the category of functors from finite dimensional F-vector spaces to K-modules, where F is a field and K is a commutative ring. We prove that, if F is a finite field, and charF is invertible in K, then the K-linear abelian category Rep(F;K) is equivalent to the product, over all n≥0, of the categories of K[GLn(F)]-modules.As a consequence, if K is also a field, then small projectives are also injective in Rep(F;K), and will have finite length. Even more is true if charK=0: the category Rep(F;K) will be semisimple.In the last section, we briefly discuss ‘q=1’ analogues and consider representations of various categories of finite sets.The main result follows from a 1992 result by L.G. Kovács about the semigroup ring K[Mn(F)].

Dualizing complexes of seminormal affine semigroup rings and toric face rings

We characterize the seminormality of an affine semigroup ring in terms of the dualizing complex, and the normality of a Cohen–Macaulay semigroup ring by the “shape” of the canonical module. We also characterize the seminormality of a toric face ring in terms of the dualizing complex. A toric face ring is a simultaneous generalization of Stanley–Reisner rings and affine semigroups.

Huneke–Wiegand conjecture and change of rings

Let R be a Cohen–Macaulay local ring of dimension one with a canonical module KR. Let I be a faithful ideal of R. We explore the problem of when I⊗RI∨ is torsionfree, where I∨=HomR(I,KR). We prove that if R has multiplicity at most 6, then I is isomorphic to R or KR as an R-module, once I⊗RI∨ is torsionfree. This result is applied to monomial ideals of numerical semigroup rings. A higher dimensional assertion is also discussed.

On the defining equations of the tangent cone of a numerical semigroup ring

Let a=a1<…<ar be a sequence of positive integers, and let Ha denote the semigroup generated by a1,…,ar. For an integer k≥0 we denote by a+k the shifted sequence a1+k,…,ar+k. Fix a field K. We show that for all k≫0 the tangent cone of the semigroup ring K[Ha+k] is Cohen–Macaulay and that it has the same Betti numbers as K[Ha+k] itself.As a consequence, we show that the number of defining equations of the tangent cone of a numerical semigroup ring is bounded by a value depending only on the width of the semigroup, where the width of a numerical semigroup is defined to be the difference of the largest and the smallest element in the minimal generating set of the semigroup. We also provide a conjectured upper bound of the above number of equations and we verify it in some cases.

The semigroup ring of a restriction semigroup with an inverse skeleton

In this paper we investigate some classes of semigroup rings with respect to (semi)primeness and (semi)primitivity. We do so by extending the techniques developed by Munn in (Proc R Soc Edinbur Sect A 107:175–196, 1987) and (Proc R Soc Edinbur Sect A 115:109–117, 1990) for the study of semigroup rings of inverse semigroups. Restriction, weakly ample and ample semigroups are considered.

A characterization of arithmetical invariants by the monoid of relations II: the monotone catenary degree and applications to semigroup rings

The investigation and classification of nonunique factorization phenomena has attracted some interest in recent literature. For finitely generated monoids, S.T. Chapman and P.A. García-Sánchez, together with several co-authors, derived a method to calculate the catenary and tame degree from the monoid of relations. Then, in Philipp (Semigroup Forum 81:424–434, 2010), the algebraic structure of this approach was investigated and the restriction to finitely generated monoids was removed. We now extend these ideas further to the monotone catenary degree and then apply all these results to the explicit computation of arithmetical invariants of semigroup rings.

A Gorenstein numerical semi-group ring having a transcendental series of Betti numbers

We prove in particular that the Gorenstein numerical semigroup ring generated by (36,48,50,52,56,60,66,67,107,121,129,135) has a transcendental series of Betti numbers. The methods of proofs are the theory of Golod homomorphism, large maps and the theory of infinite positively graded Lie algebras.

Depth of Initial Ideals of Normal Edge Rings

Let G be a finite graph on the vertex set [d] = {1,…, d} with the edges e 1,…, e n and K[t] = K[t 1,…, t d ] the polynomial ring in d variables over a field K. The edge ring of G is the semigroup ring K[G] which is generated by those monomials t e = t i t j such that e = {i, j} is an edge of G. Let K[x] = K[x 1,…, x n ] be the polynomial ring in n variables over K, and define the surjective homomorphism π: K[x] → K[G] by setting π(x i ) = t e i for i = 1,…, n. The toric ideal I G of G is the kernel of π. It will be proved that, given integers f and d with 6 ≤ f ≤ d, there exists a finite connected nonbipartite graph G on [d] together with a reverse lexicographic order <rev on K[x] and a lexicographic order <lex on K[x] such that (i) K[G] is normal with Krull-dim K[G] = d, (ii) depth K[x]/in<rev (I G ) = f and K[x]/in<lex (I G ) is Cohen–Macaulay, where in<rev (I G ) (resp., in<lex (I G )) is the initial ideal of I G with respect to <rev (resp., <lex) and where depth K[x]/in<rev (I G ) is the depth of K[x]/in<rev (I G ).

Direct limits of Cohen–Macaulay rings

Let A be a direct limit of a direct system of Cohen–Macaulay rings. In this paper, we describe the Cohen–Macaulay property of A. Our results indicate that A is not necessarily Cohen–Macaulay. We show A is Cohen–Macaulay under various assumptions. As an application, we study Cohen–Macaulayness of non-affine normal semigroup rings.

On free resolutions of some semigroup rings

For some numerical semigroup rings of small embedding dimension, namely those of embedding dimension 3, and symmetric or pseudosymmetric of embedding dimension 4, presentations has been determined in the literature. We extend these results by giving the whole graded minimal free resolutions explicitly. Then we use these resolutions to determine some invariants of the semigroups and certain interesting relations among them. Finally, we determine semigroups of small embedding dimensions which have strongly indispensable resolutions.