Prime Ideals in Algebras Determined by Submonoids of Nilpotent Groups

The prime spectrum of the semigroup algebra K[S] of a submonoid S of a finitely generated nilpotent group is studied via the spectra of the monoid S and of the group algebra K[G] of the group G of fractions of S. It is shown that the classical Krull dimension of K[S] is equal to the Hirsch length of the group G provided that G is nilpotent of class two. This uses the fact that prime ideals of S are completely prime. An infinite family of prime ideals of a submonoid of a free nilpotent group of class three with two generators which are not completely prime is constructed. They lead to prime ideals of the corresponding algebra. Prime ideals of the monoid of all upper triangular n × n matrices with non-negative integer entries are described and it follows that they are completely prime and finite in number.     

Submonoids of Polycyclic-by-Finite Groups and their Algebras

We describe Noetherian semigroup algebras K[S] of submonoids S of polycyclic-by-finite groups over a field K. As an application, we show that these algebras are finitely presented and also that they are Jacobson rings. Next we show that every prime ideal P of K[S] is strongly related to a prime ideal of the group algebra of a subgroup of the quotient group of S via a generalised matrix ring structure on K[S]/P. Applications to the classical Krull dimension, prime spectrum, and irreducible K[S]-modules are given.     

Computing invariants of algebraic groups in arbitrary characteristic

Let G be an affine algebraic group acting on an affine variety X. We present an algorithm for computing generators of the invariant ring KG[X] in the case where G is reductive. Furthermore, we address the case where G is connected and unipotent, so the invariant ring need not be finitely generated. For this case, we develop an algorithm which computes KG[X] in terms of a so-called colon-operation. From this, generators of KG[X] can be obtained in finite time if it is finitely generated. Under the additional hypothesis that K[X] is factorial, we present an algorithm that finds a quasi-affine variety whose coordinate ring is KG[X]. Along the way, we develop some techniques for dealing with nonfinitely generated algebras. In particular, we introduce the finite generation ideal.     

Injective Hulls of Simple Modules over Differential Operator Rings

We study injective hulls of simple modules over differential operator rings R[θ; d], providing necessary conditions under which these modules are locally Artinian. As a consequence, we characterize Ore extensions of S = K[x][θ; σ, d] for σ a K-linear automorphism and d a K-linear σ-derivation of K[x] such that injective hulls of simple S-modules are locally Artinian.     

Serre's Condition Rℓ for Affine Semigroup Rings

In this note we characterize the affine semigroup rings K[S] over an arbitrary field K that satisfy condition R_? of Serre. Our characterization is in terms of the face lattice of the positive cone pos(S) of S. We start by reviewing some basic facts about the faces of pos(S) and consequences for the monomial primes of K[S]. After proving our characterization we turn our attention to the Rees algebras of a special class of monomial ideals in a polynomial ring over a field. In this special case, some of the characterizing criteria are always satisfied. We give examples of non-normal affine semigroup rings that satisfy R₂.     

Some Examples of Affine Monoid Algebras

We construct a finitely generated monoid S with a zero element such that for every field K the Jacobson radical of the monoid algebra K[S] is a sum of nilpotent ideals but is not nilpotent. Moreover, the contracted monoid algebra K ₀[S] is a monomial algebra.

If K is a field of characteristic p > 0, then we construct a finitely presented group H _p such that the Jacobson radical J of the group algebra K[H _p ] is a sum of nilpotent ideals, but is not nilpotent. Moreover, K[H _p ]/J is a domain.     

NOETHERIAN UNIQUE FACTORIZATION SEMIGROUP ALGEBRAS

We investigate when semigroup algebras K[S] of submonoids S of torsion free polycyclic-by-finite groups G are Noetherian unique factorization rings in the sense of Chatters and Jordan, that is, every prime ideal contains a principal height one prime ideal. For the group algebra K[G] this problem was solved by Brown.     

Monomial modules and graded Betti numbers

Let K be a field, S = K[x ₁,…, x _n ], the polynomial ring over K, and let F be a finitely generated graded free S-module with homogeneous basis. Certain invariants, such as the Castelnuovo-Mumford regularity and the graded Betti numbers of submodules of F, are studied; in particular, the componentwise linear submodules of F are characterized in terms of their graded Betti numbers.     

Note on bounds for multiplicities

Let S=K[x₁,…,x_n] be a polynomial ring and R=S/I be a graded K-algebra where I⊂S is a graded ideal. Herzog, Huneke and Srinivasan have conjectured that the multiplicity of R is bounded above by a function of the maximal shifts in the minimal graded free resolution of R over S. We prove the conjecture in the case that codim(R)=2 which generalizes results in (J. Pure Appl. Algebra 182 (2003) 201; Trans. Amer. Math. Soc. 350 (1998) 2879). We also give a proof for the bound in the case in which I is componentwise linear. For example, stable and squarefree stable ideals belong to this class of ideals.     

Wedderburn polynomials over division rings, I

A Wedderburn polynomial over a division ring K is a minimal polynomial of an algebraic subset of K. Such a polynomial is always a product of linear factors over K, although not every product of linear polynomials is a Wedderburn polynomial. In this paper, we establish various properties and characterizations of Wedderburn polynomials over K, and show that these polynomials form a complete modular lattice that is dual to the lattice of full algebraic subsets of K. Throughout the paper, we work in the general setting of an Ore skew polynomial ring K[t,S,D], where S is an endomorphism of K and D is an S-derivation on K.     

On the polynomial closure in a valued field

Let D be an integral domain with quotient field K. For any subset S of K, the D-polynomial closure of S is the largest subset T of K such that, for every polynomial f in K[X], if f maps S into D then f maps also T into D. When D is not local, the D-polynomial closure is not a topological closure. We prove here that, when D is any rank-one valuation domain, then there exists a topology on K such that the closed subsets for this topology are exactly the D-polynomially closed subsets of K.     

Embedding Witt rings of Dedekind domains

Let W(R) denote Harrison's Witt ring of the commutative ring R. In case R is a field of characteristic ≠ 2, this is the classical Witt ring based on anisotropic quadratic forms. In this note we determine under what conditions W(R) is embedded in W(S) for certain Dedekind domains R ? S. In particular, an answer is given in case R and S are the integers in algebraic number fields K and L, respectively, with (L: K) odd.     

An analogue of group divisible designs for Moore graphs

We study graphs whose adjacency matrix S of order n satisfies the equation S + S² = J ? K + kI, where J is a matrix of order n of all 1's, K is the direct sum on $\text{n}\text{l}$ matrices of order l of all 1's, and I is the identity matrix. Moore graphs are the only solutions to the equation in the case l = 1 for which K = I. In the case k = l we can obtain Moore graphs from a solution S by a bordering process analogous to obtaining (ν, κ, λ)-designs from some group divisible designs. Other parameters are rare. We are able to find one new interesting graph with parameters k = 6, l = 4 on n = 40 vertices. We show that it has a transitive automorphism group isomorphic to C₄ × S₅.     

Prime Ideals of Cancellative Semigroups

Abstract

For a class of prime ideals P of a cancellative semigroup S it is shown that the factors S/P have a structure of a monomial semigroup over a group. Consequences for the semigroup algebras K[S] are discussed.     

Image of the group ring of the Galois representation associated to Drinfeld modules

Let φ be a Drinfeld A-module of arbitrary rank and arbitrary characteristic over a finitely generated field K, and set GK=Gal(Ksep/K). Let E=EndK(φ). We show that for almost all primes p of A the image of the group ring A[GK] in EndA(Tp(φ)) is the commutant of E. In the special case E=A it follows that the representation of GK on the p-torsion points φ[p](Ksep) of φ is absolutely irreducible for almost all p.     

Von Neumann finite endomorphism rings

Let K be a field of characteristic zero, G a group acting on a nonempty set X and KX the permutation module induced by this action. By studying traces of idempotents, we prove that the endomorphism ring End_K[G](KX) is von Neumann finite under certain conditions for the action of G on X. This generalizes a classical result by Kaplansky for the group ring of G over K.     

Tandem Queues with Impatient Customers for Blood Screening Procedures

We study a blood testing procedure for detecting viruses like HIV, HBV and HCV. In this procedure, blood samples go through two screening steps. The first test is ELISA (antibody Enzyme Linked Immuno-Sorbent Assay). The portions of blood which are found not contaminated in this first phase are tested in groups through PCR (Polymerase Chain Reaction). The ELISA test is less sensitive than the PCR test and the PCR tests are considerably more expensive. We model the two test phases of blood samples as services in two queues in series; service in the second queue is in batches, as PCR tests are done in groups. The fact that blood can only be used for transfusions until a certain expiration date leads, in the tandem queue, to the feature of customer impatience. Since the first queue basically is an infinite server queue, we mainly focus on the second queue, which in its most general form is an S-server M/G ^[k,?K]/S?+?G queue, with batches of sizes which are bounded by k and K. Our objective is to maximize the expected profit of the system, which is composed of the amount earned for items which pass the test (and before their patience runs out), minus costs. This is done by an appropriate choice of the decision variables, namely, the batch sizes and the number of servers at the second service station. As will be seen, even the simplest version of the batch queue, the M/M ^[k,?K]/1?+?M queue, already gives rise to serious analytical complications for any batch size larger than 1. These complications are discussed in detail, and handled for K?=?2. In view of the fact that we aim to solve realistic optimization problems for blood screening procedures, these analytical complications force us to take recourse to either a numerical approach or approximations. We present a numerical solution for the queue length distribution in the M/M ^[k,?K]/S?+?M queue and then formulate and solve several optimization problems. The power-series algorithm, which is a numerical-analytic method, is also discussed.     

A complete parametrization of cyclic field extensions of 2-power degree

Letq be a power of 2 at least equal to 8 and ζ be a primitiveq-th root of unity, and letK be any field of characteristic zero. We define the group of special projective conormsS _K as a quotient of the group of elements ofK(ζ) of norm 1:S _K is obviously trival if the groul Gal (K(ζ)/K) is cyclic. We prove that for some fieldsK, the groupS _K is finite, and it is even trivial for certain fields such as ? or ?(X ₁,...,X _m). We then prove that the groupS _K completely paramatrizes the cycle extensions ofK of degreeq. We exhibit an explicit polynomial defined over ?(T ₀,...,T _q/2) which parametrizes all cyclic extensions ofK of degreeq associated to the trivial element ofS _K. In particular, this polynomial parametrizes all cyclic extensions ofK of degreeq whenever the groupS _K is trivial.     

Noetherian Semigroup Algebras

A complete structural characterization of submonoids S of apolycyclic-by-finite group such that the semigroup algebra K[S]over a field K is right noetherian is obtained. It follows thatsuch algebras are also left noetherian. 2000 Mathematics SubjectClassification 16P40, 16S36, 20M25 (primary), 20F22, 20C07,20M10 (secondary).     

Security in graphs

Let G=(V,E) be a graph. A set S⊆V is a defensive alliance if |N[x]∩S|?|N[x]-S| for every x∈S. Thus, each vertex of a defensive alliance can, with the aid of its neighbors in S, be defended from attack by its neighbors outside of S. An entire set S is secure if any subset X⊆S can be defended from an attack from outside of S, under an appropriate definition of what such a defense implies. Necessary and sufficient conditions for a set to be secure are determined.