Remarks on blowing-up divisorial ideals

In this short paper I describe special situations where a local ring is normally flat along a divisorial prime ideal.This work was supported by C.N.R. (Consiglio Nazionale delle Ricerche)     

Note on some ideals of associative rings

It is well known that being an ideal (left ideal, right ideal) of a ring is not a transitive relation. Nevertheless in some cases the transitive property does hold. Systematic studies of this subject were started in [11,13]. In this paper we continue these studies.     

Note on generalized open sets

Characterizations of γ-open sets and locally γ-regular sets are given. We generalize some already established results and answer an open question by giving a characterization to γ-quasi-open sets.      

Canonical characteristic sets of characterizable differential ideals

Canonical characteristic sets of characterizable differential ideals are studied. For the ordinary case, an estimate of orders of elements from the canonical characteristic set is proved. It is also shown how one can verify the equality of characterizable ideals without using canonical characteristic sets.     

On divisorial filtrations on sheaves

In this paper, we generalize the notion of Poincaré series of a multi-index divisorial filtration corresponding to a collection of sigma-processes to the case of an arbitrary locally-free sheaf on the space of blow-ups of the complex plane ?². For an arbitrary sheaf, we establish a representation of the series in terms of topological invariants of the sheaf. In particular, for the sheaf of functions, this representation coincides with the Poincaré series obtained by Gusein—Zade and Delgado.     

Prime elements from prime ideals

The prime ideal theorem for distributive lattices (PIT) is shown to imply that any complete distributive lattice with a compact unit has a prime element, which is then used to deduce from PIT that (1) every nontrivial ring with unit has a prime ideal, and (2) every Wallman locale is spatial.     

Asymptotic behaviour of certain sets of prime ideals

Let be ideals of the commutative ring , let be a Noetherian -module and let be a submodule of ; also let be an Artinian -module and let be a submodule of . It is shown that, whenever is a sequence of -tuples of non-negative integers which is non-decreasing in the sense that for all and all , then Ass is independent of for all large , and also Att is independent of for all large . These results are proved without any regularity conditions on the ideals , and so (a special case of) the first answers in the affirmative a question raised by S. McAdam.

     

Note on the number of integral ideals in Galois extensions

Let K be an algebraic number field of finite degree over the rational filed Q.Let ak be the number of integral ideals in K with norm k.In this paper we study the l-th integral power sum of ak,i.e.,∑k≤ x akl(l = 2,3,...).We are able to improve the classical result of Chandrasekharan and Good.As an application we consider the number of solutions of polynomial congruences.     

Free sets and the structure of additive ideals

Summary The present paper contains existence theorems for free sets with respect to a set-mapping. Paper done at the ? Istituto Nazionale per la Applicazioni del Calcolo ? of Rome during the stay of Dr.G. Fodor, with a fellowship of the Italian Governement.     

Note on independent sets in steiner systems

A partial Steiner (n, k, l)-system or briefly (n, k, l)-system is a pair (V, S), where V is an n-set and S is a collection of k-subsets of V, such that every l-subset of V is contained in at most one k-subset of S. A subset X ? V is called independent if [X]^k ∩ S = 0. The size of the largest independent set in S is denoted by α(S). Define The purpose of this note is to prove that for every k, l, k > l holds, where c, d are positive constants depending on k and l only.     

Note on difference sets in Z n

We give a purely elementary-combinatorial proof of a result of Bergelson which was derived from an ergodic theorem This revised version was published online in June 2006 with corrections to the Cover Date.     

Representations of positive polynomials on noncompact semialgebraic sets via KKT ideals

This paper studies the representation of a positive polynomial f(x) on a noncompact semialgebraic set S={x∈Rⁿ:g₁(x)≥0,…,g_s(x)≥0} modulo its KKT (Karush-Kuhn-Tucker) ideal. Under the assumption that the minimum value of f(x) on S is attained at some KKT point, we show that f(x) can be represented as sum of squares (SOS) of polynomials modulo the KKT ideal if f(x)>0 on S; furthermore, when the KKT ideal is radical, we argue that f(x) can be represented as a sum of squares (SOS) of polynomials modulo the KKT ideal if f(x)≥0 on S. This is a generalization of results in [J. Nie, J. Demmel, B. Sturmfels, Minimizing polynomials via sum of squares over the gradient ideal, Mathematical Programming (in press)], which discusses the SOS representations of nonnegative polynomials over gradient ideals.     

Minimal sets of generators for the relation ideals of certain monomial curves

LetO be a monomial curve in the affine algebraice-space over a fieldK andP be the relation ideal ofO. ifO is defined by a sequence ofe positive integers somee-1 of which form an arithmetic sequence then we construct a minimal set of generators forP and write an explicit formula for μ(P).     

Note on prime representations of convex polyhedral sets

Consider a convex polyhedral set represented by a system of linear inequalities. A prime representation of the polyhedron is one that contains no redundant constraints. We present a sharp upper bound on the difference between the cardinalities of any two primes.This research was supported by the Natural Sciences and Engineering Research Council of Canada under Grant Nos. A8807, A4625, and A7742.