On the projectivity of proper normal curves over valuation domains

A theorem of Lichtenbaum states, that every proper, regular curve [Formula: see text] over a discrete valuation domain [Formula: see text] is projective. This theorem is generalized to the case of an arbitrary valuation domain [Formula: see text] using the following notion of regularity for non-noetherian rings introduced by Bertin: the local ring [Formula: see text] of a point [Formula: see text] is called regular, if every finitely generated ideal [Formula: see text] has finite projective dimension. The generalization is a particular case of a projectivity criterion for proper, normal [Formula: see text]-curves: such a curve [Formula: see text] is projective if for every irreducible component [Formula: see text] of its closed fiber [Formula: see text] there exists a closed point [Formula: see text] of the generic fiber of [Formula: see text] such that the Zariski closure [Formula: see text] meets [Formula: see text] and meets [Formula: see text] in regular points only.

Descending chains of semistar operations

A class of integer-valued functions defined on the set of ideals of an integral domain [Formula: see text] is investigated. We show that this class of functions, which we call ideal valuations, are in one-to-one correspondence with countable descending chains of finite type, stable semistar operations with largest element equal to the [Formula: see text]-operation. We use this class of functions to recover familiar semistar operations such as the [Formula: see text]-operation and to give a solution to a conjecture by Chapman and Glaz when the ring is a valuation domain.

A Finitely Generated Module over a Valuation Domain

This article discusses about some properties which are equivalent between a finitely generated module over PID and a finitely generated module over a valuation domain. This can be done by considering a finitely generated module over a DVR. Although in general a PID is not a valuation domain or vice versa, these equivalence of some properties will be valid. It is because a DVR is a PID and a valuation domain at the same time. Those the equivalent properties in a finitely generated module over DVR are related with the decomposition of the module and the height of an element in that module.<strong></strong>

Bhargava rings that are Prüfer v-multiplication domains

We study the P[Formula: see text]MD property of Bhargava rings [Formula: see text]. In particular, we explore the cases in which [Formula: see text] is a valuation domain, a Krull-type domain, or an almost Dedekind domain.

Intersection of Ideals in a Polynomial Ring over a Dual Valuation Domain

Let V be a valuation domain and let A=V+εV be a dual valuation domain. We propose a method for computing a strong Gröbner basis in R=A[x1,…,xn]; given polynomials f1,…,fs∈R, a method for computing a generating set for Syz(f1,…,fs)={(h1,…,hs)∈Rs∣h1f1+⋯+hsfs=0} is given; and, finally, given two ideals I=〈f1,…,fs〉 and J=〈g1,…,gr〉 of R, we propose an algorithm for computing a generating set for I∩J.