Star Operations on Numerical Semigroups

It is proved that the number of numerical semigroups with a fixed number n of star operations is finite if n > 1. The result is then extended to the class of analytically irreducible residually rational one-dimensional Noetherian rings with finite residue field and integral closure equal to a fixed discrete valuation domain.     

Polynomes a valeurs entieres sur un anneau de pseudovaluation

We study the ring of integral valued polynomials over a pseudovaluation domain A. We entirely determine the set of prime ideals above the maximal ideal M of A: if M is a principal ideal in the valuation domain V associated with A and if its residue field is finite, then this set is in bijection with a topologically complete ring, as in the Noetherian case; if M is principal but of infinite residue field in V, then this set is finite; at last, if M is not principal, then the ring of integral valued polynomials is included in V[X] and has the same set of prime ideals above M.     

Dimension de l’anneau des polynomes a valeurs entieres

Let A be a commutative domain with quotient field K and A_S the ring of integer-valued polynomials thus A_S={f∈K[X]; f(A)⊂A}; we show that the Krull dimension of A_S is such that dim A_S≥dim A[X]-1 and give examples where dim A_S=dim A[X]-1. Considering chains of primes of A_S above a maximal idealm of finite residue field, we give also examples where the length of such a chain can arbitrarily be prescribed (whereas in A[X] the length of such chains is always 1). To provide such examples we consider a pair of domains A⊂B sharing an ideal I such that A/I is finite; we give sufficient conditients to have A_S⊂B[X] and show that in this case dim A_S=dim B[X]. At last, as a generalisation of Noetherian rings of dimension 1, we consider domains with an ideal I such that A/I is finite and a power Iⁿ of I is contained in a proper principal ideal of A; for such domains we show that every prime of A_S above a primem containing I is maximal.      

Separable polynomials over finite dimensional algebras

In [5] it was shown that for a polynomial P of precise degree n with coefficients in an arbitrary m-ary algebra of dimension d as a vector space over an algebraically closed fields, the zeros of P together with the homogeneous zeros of the dominant part of P form a set of cardinality n^d or the cardinality of the base field. We investigate polynomials with coefficients in a d dimensional algebra A without assuming the base field k to be algebraically closed. Separable polynomials are defined to be those which have exactly n^d distinct zeros in [Ktilde] ?_k A [Ktilde] where [Ktilde] denotes an algebraic closure of k. The main result states that given a separable polynomial of degree n, the field extension L of minimal degree over k for which L ?_k A contains all nd zeros is finite Galois over k. It is shown that there is a non empty Zariski open subset in the affine space of all d-dimensional k algebras whose elements A have the following property: In the affine space of polynomials of precise degree n with coefficients in A there is a non empty Zariski open subset consisting of separable polynomials; in other polynomials with coefficients in a finite dimensional algebra are “generically” separable.     

Overrings of two-dimensional Noetherian domains representable by Noetherian spaces of valuation rings

Let D be a Noetherian domain of Krull dimension 2, and let H⊆R be integrally closed overrings of D. We examine when H can be represented in the form H=(?_V∈ΣV)∩R, with Σ a Noetherian subspace of the Zariski-Riemann space of the quotient field of D. We characterize also the special case in which Σ can be chosen to be a finite character collection of valuation overrings of D.     

Noetherian rings with almost injective simple modules

We characterize right Noetherian rings over which all simple modules are almost injective. It is proved that R is such a ring, if and only if, the complements of semisimple submodules of every R-module M are direct summands of M, if and only if, R is a finite direct sum of right ideals I_r, where I_r is either a Noetherian V-module with zero socle, or a simple module, or an injective module of length 2. A commutative Noetherian ring for which all simple modules are almost injective is precisely a finite direct product of rings R_i, where R_i is either a field or a quasi-Frobenius ring of length 2. We show that for commutative rings whose all simple modules are almost injective, the properties of Kasch, (semi)perfect, semilocal, quasi-Frobenius, Artinian, and Noetherian coincide.     

THE NUMBER AND SIZE OF ORBITS OF SOME PERMUTATION GROUPS

Abstract

It is shown that Aut ?, the group of homeomorphisms of the rational numbers with the usual topology, has 2 ^N_o orbits on the power set P(?). We call S ? ? a moiety if S and its complement in ? are infinite. It is shown that the orbit of any moiety S under Aut ? has cardinality 2^N_o while the orbit of S under Aut(?, ≤), the group of order preserving automorphisms of ?, has cardinality N_o if and only if S is a finite union of disjoint rational intervals with rational endpoints.     

Chain and distributive coalgebras

We show that coalgebras whose lattice of right coideals is distributive are coproducts of coalgebras whose lattice of right coideals is a chain. Those chain coalgebras are characterized as finite duals of Noetherian chain rings whose residue field is a finite dimensional division algebra over the base field. They also turn out to be coreflexive. Infinite dimensional chain coalgebras are finite duals of left Noetherian chain domains. Given any finite dimensional division algebra D and D-bimodule structure on D, we construct a chain coalgebra as a cotensor coalgebra. Moreover if D is separable over the base field, every chain coalgebra of type D can be embedded in such a cotensor coalgebra. As a consequence, cotensor coalgebras arising in this way are the only infinite dimensional chain coalgebras over perfect fields. Finite duals of power series rings with coeficients in a finite dimensional division algebra D are further examples of chain coalgebras, which also can be seen as tensor products of D^∗, and the divided power coalgebra and can be realized as the generalized path coalgebra of a loop. If D is central, any chain coalgebra is a subcoalgebra of the finite dual of D[[x]].     

Sur le theoreme de Stone-Weierstrass en Algebre commutative

We show that the sufficient conditions given by Cahen, Grazzini and Haouat for a version of the Stone-Weierstrass theorem in commutative algebra are the widest. More precisely, letA be a Noetherian ring andI a proper ideal ofA such thatA is Hausdorff with respect to theI-adic topology. Note the completion ofA andC(Â,Â) the ring of continuous functions from to with uniform convergence topology. The subset of polynomial functions is dense inC(Â,Â) if and only if the radical ofI is a maximal idealm ofA and the local ringA _m is a one-dimensional analytically irreducible domain with finite residue field.     

On the divisorial spectrum of a Noetherian domain

For a Noetherian domain, the sets of divisorial primes, t-primes, and associated primes of principal ideals coincide. We study the divisorial primes of a Noetherian domain as a partially ordered set. In particular, we show that it is possible to have arbitrarily long chains and any finite amount of noncatenarity.     

Flat Modules over Group Rings of Finite Groups

Let k be a commutative ring of coefficients and G be a finite group. Does there exist a flat k G-module which is projective as a k-module but not as a k G-module? We relate this question to the question of existence of a k-module which is flat and periodic but not projective. For either question to have a positive answer, it is at least necessary to have |k| ≥ ?_ω. There can be no such example if k is Noetherian of finite Krull dimension, or if k is perfect.     

A tower with non-Galois steps which attains the Drinfeld-Vladut bound

We introduce a new tower of function fields over a finite field of square cardinality, which attains the Drinfeld-Vladut bound. One new feature of this new tower is that it is constructed with non-Galois steps; i.e., with non-Galois function field extensions. The exact value of the genus g(F_n) is also given (see Lemma 4).     

Finiteness and Skolem Closure of Ideals for Non-Unibranched Domains

A one-dimensional, Noetherian, local domain D with maximal ideal 𝔪 and finite residue field was known to be an almost strong Skolem ring if analytically irreducible. It was unknown whether this condition is necessary. We show that it is at least necessary for D to be unibranched. After introducing a general notion of equalizing ideal, we show that, for k large enough, the ideals of the form 𝔐 _{k, a}  = {f ∈ Int(D) | f(a) ∈ 𝔪 ^k }, for a ∈ D, are distinct. This allows to show that the maximal ideals 𝔐 _a  = {f ∈ Int(D) | f(a) ∈ 𝔪}, although not necessarily distinct, are never finitely generated.     

Direct sums of injective semimodules and direct products of projective semimodules over semirings

We prove that if the direct sum of a family of semimodules over a semiring S is an injective semimodule or if the direct product of a family of semimodules over S is a projective semimodule, then the cardinality of the subfamily consisting of all semimodules which are not modules is strictly less than the cardinality of S. As a consequence, we obtain semiring analogs of well-known characterizations of classical semisimple, quasi-Frobenius, and one-sided Noetherian rings.     

Finite orbits for rational functions

Let K be a number field and φ ∈ K(z) a rational function. Let S be the set of all archimedean places of K and all non-archimedean places associated to the prime ideals of bad reduction for φ. We prove an upper bound for the length of finite orbits for φ in ?₁ (K) depending only on the cardinality of S.     

Uppers to zero in intersections of prime ideals

Let R be a Noetherian integral domain. The structure of the partially-ordered set of prime ideals of R[z], the polynomial ring in one indeterminate over R, is not fully understood. I demonstrate that if p₁,…,p_n are prime ideals in R[x] with ht(p_i) > 2 and either n = 1 or R is not a Henselian local domain of dimension < 2, then pi D-o-C\pn contains [R] many prime ideals which intersect R at (0). I also show that if R is a Noetherian domain that is not a Henselian local domain and p₁,…,p_n are prime ideals with height > 2 each of which contains a monic polynomial, then their intersection contains [R] many prime ideals meeting R at (0), each containing a monic polynomial.     

Selforthogonal Modules with Finite Injective Dimension III

Let R be a left Noetherian ring, S a right Noetherian ring and _R U a generalized tilting module with S?=?End( _R U). We give some equivalent conditions that the injective dimension of U _S is finite implies that of _R U is also finite. As an application, under the assumption that the injective dimensions of _R U and U _S are finite, we construct a hereditary and complete cotorsion theory by some subcategories associated with _R U.     

Projective character degree patterns of 2-groups

Abstract

We discuss the prospects for finding a “core class,” i.e., a well-behaved class of non-free abelian groups of cardinality ?₁ such that every non-free abelian group of cardinality ?₁ has a subgroup in the core class.     

Agreeable domains

An integral domain D with quotient field K is defined to be agreeable if for each fractional ideal F of D[X] with F C K[X] there exists 0 = s ε D with sF C D[X]. D is agreeable ? D satisfies property (*) (for 0 ^ f(X) G K[X], there exists 0 = s ε D so that f(X)g(X) ε D[X] for g(X) ε K[X] implies that sg(X) ε D[X]) &; D[X] is an almost principal domain, i.e., for each nonzero ideal I of D[X] with IK[X] = K[X], there exists f(X) ε I and 0 = s ε D with sI C (f(X)). If D is Noetherian or integrally closed, then D is agreeable. A number of other characterizations of agreeable domains are given as are a number of stability properties. For example, if D is agreeable, so is ?^αD_{P α} and for a pair of domains D?D′ with a [DD:′]≠0, D is agreeable?D′ is agreeable. Results on agreeable domains are used to give an alternative treatment of Querre's characterization of divisorial ideals in integrally closed polynomial rings. Finally, the various characterizations of D being agreeable are considered for polynomial rings in several variables.     

Generators and Dimensions of Derived Categories of Modules

Several years ago, Bondal, Rouquier, and Van den Bergh introduced the notion of the dimension of a triangulated category, and Rouquier proved that the bounded derived category of coherent sheaves on a separated scheme of finite type over a perfect field has finite dimension. In this article, we study the dimension of the bounded derived category of finitely generated modules over a commutative Noetherian ring. The main result of this article asserts that it is finite over a complete local ring containing a field with perfect residue field. Our methods also give a ring-theoretic proof of the affine case of Rouquier's theorem.