We present an algorithm to compute a primary decomposition of an ideal in a polynomial ring over the integers. For this purpose we use algorithms for primary decomposition in polynomial rings over the rationals resp. over finite fields, and the idea of Shimoyama–Yokoyama resp. Eisenbud– Hunecke–Vasconcelos to extract primary ideals from pseudo–primary ideals. A parallelized version of the algor...

We give a characterization of primary ideals of finitely generated commutative monoids and in the case of finitely generated cancellative monoids we give an algorithmic method for deciding if an ideal is primary or not. Finally we give some properties of primary elements of a cancellative monoid and an algorithmic method for determining the primary elements of a finitely generated cancellative ...

We propose a candidate, which we call the fractional Galois ideal after Snaith’s fractional ideal, for replacing the classical Stickelberger ideal associated to an abelian extension of number fields. The Stickelberger ideal can be seen as gathering information about those L-functions of the extension which are non-zero at the special point s = 0, and was conjectured by Brumer to give annihilato...

Let A be a Noetherian local ring with the maximal ideal m and an m-primary ideal J . Let F = {In}n≥0 be a good filtration of ideals in A. Denote by FJ (F) = ⊕ n≥0 (In/JIn)t n the fiber cone of F with respect to J. The paper characterizes the multiplicity and the CohenMacaulayness of FJ (F) in terms of minimal reductions of F .

In this paper we deal with (ideal) continuity of lattice group-valued finitely additive measures, and prove some basic properties and comparison results. We investigate the relations between different modes of ideal continuity, and give some characterization. Finally we pose some open problems. Mathematics Subject Classification: Primary: 28A12, 28A33, 28B10, 28B15, 28C10, 40A35, 46G10. Seconda...

Let R be a local, Noetherian ring and I ⊆ R an ideal. A question of Kodiyalam asks whether for fixed i > 0, the polynomial giving the ith Betti number of In has degree equal to the analytic spread of I minus one. Under mild conditions on R, we show that the answer is positive in a number of cases, including when I is divisible by m or I is an integrally closed m-primary ideal.

By employing the (de)homogenization technique in a relatively extensive setting, this note studies in detail the relation between nonhomogeneous Gröbner bases and homogeneous Gröbner bases. As a consequence, a general principle of computing Gröbner bases (for an ideal and its homogenization ideal) by passing to homogenized generators is clarified systematically. The obtained results improve and...

Let (R, m) be a Cohen-Macaulay local ring of dimension d > 0, I an m-primary ideal of R and K an ideal containing I. When depth G(I) ≥ d− 1 and r(I|K) < ∞, we present an upper bound on the second fiber coefficient f2(I, K) of the fiber cones FK(I), and also provide a characterization, in terms of f2(I, K), of the condition depth FK(I) ≥ d− 2.