Let and be ideals of a Noetherian local ring and let be a nonzero finitely generated -module. We study the relation between the vanishing of and the comparison of certain ideal topologies. Also, we characterize when the integral closure of an ideal relative to the Noetherian -module is equal to its integral closure relative to the Artinian -module .
Let (U) denote the algebra of holomorphic functions on an open subset U ⊂ ℂⁿ and Z ⊂ (U) its finite-dimensional vector subspace. By the theory of least spaces of de Boor and Ron, there exists a projection from the local ring onto the space of germs of elements of Z at b. At a general point b ∈ U its kernel is an ideal and induces the structure of an Artinian algebra on . In particular, this holds at points where the kth jets of elements of Z form a vector bundle for each k ∈ ℕ. For an embedded...
On the ring of polynomials in n variables over a field special isomorphisms ’s of into are defined which preserve the greatest common divisor of two polynomials. The ring is extended to the ring and the ring of generalized polynomials in such a way that the exponents of the variables are non-negative rational numbers and rational numbers, respectively. The isomorphisms ’s are extended to automorphisms ’s of the ring . Using the property that the isomorphisms ’s preserve GCD it...
The notion of a real semigroup was introduced in [8] to provide a framework for the investigation of the theory of (diagonal) quadratic forms over commutative, unitary, semi-real rings. In this paper we introduce and study an outstanding class of such structures, that we call spectral real semigroups (SRS). Our main results are: (i) The existence of a natural functorial duality between the category of SRSs and that of hereditarily normal spectral spaces; (ii) Characterization of the SRSs as the...