Krull modules and completely integrally closed modules

Let [Formula: see text] be a torsion-free module over an integral domain [Formula: see text] with quotient field [Formula: see text]. We define a concept of completely integrally closed modules in order to study Krull modules. It is shown that a Krull module [Formula: see text] is a [Formula: see text]-multiplication module if and only if [Formula: see text] is a maximal [Formula: see text]-submodule and [Formula: see text] for every minimal prime ideal [Formula: see text] of [Formula: see text]. If [Formula: see text] is a finitely generated Krull module, then [Formula: see text] is a Krull module and [Formula: see text]-multiplication module. It is also shown that the following three conditions are equivalent: [Formula: see text] is completely integrally closed, [Formula: see text] is completely integrally closed, and [Formula: see text] is completely integrally closed.

$$v_c$$-Noetherian domains and Krull domains

Let D be an integrally closed domain, $$\{V_{\alpha }\}$$ { V α } be the set of t-linked valuation overrings of D, and $$v_c$$ v c be the star operation on D defined by $$I^{v_c} = \bigcap _{\alpha } IV_{\alpha }$$ I v c = ⋂ α I V α for all nonzero fractional ideals I of D. In this paper, among other things, we prove that D is a $$v_c$$ v c -Noetherian domain if and only if D is a Krull domain, if and only if $$v_c = v$$ v c = v and every prime t-ideal of D is a maximal t-ideal. As a corollary, we have that if D is one-dimensional, then $$v_c = v$$ v c = v if and only if D is a Dedekind domain.

FCP $$\Delta $$-extensions of rings

We consider ring extensions, whose set of all subextensions is stable under the formation of sums, the so-called $$\Delta $$ Δ -extensions. An integrally closed extension has the $$\Delta $$ Δ -property if and only it is a Prüfer extension. We then give characterizations of FCP $$\Delta $$ Δ -extensions, using the fact that for FCP extensions, it is enough to consider integral FCP extensions. We are able to give substantial results. In particular, our work can be applied to extensions of number field orders because they have the FCP property.

GENERALISED QUANTUM DETERMINANTAL RINGS ARE MAXIMAL ORDERS

Generalised quantum determinantal rings are the analogue in quantum matrices of Schubert varieties. Maximal orders are the noncommutative version of integrally closed rings. In this paper, we show that generalised quantum determinantal rings are maximal orders. The cornerstone of the proof is a description of generalised quantum determinantal rings, up to a localisation, as skew polynomial extensions.

On almost valuation ring pairs

If [Formula: see text] are (commutative) rings, [Formula: see text] denotes the set of intermediate rings and [Formula: see text] is called an almost valuation (AV)-ring pair if each element of [Formula: see text] is an AV-ring. Many results on AV-domains and their pairs are generalized to the ring-theoretic setting. Let [Formula: see text] be rings, with [Formula: see text] denoting the integral closure of [Formula: see text] in [Formula: see text]. Then [Formula: see text] is an AV-ring pair if and only if both [Formula: see text] and [Formula: see text] are AV-ring pairs. Characterizations are given for the AV-ring pairs arising from integrally closed (respectively, integral; respectively, minimal) ring extensions [Formula: see text]. If [Formula: see text] is an AV-ring pair, then [Formula: see text] is a P-extension. The AV-ring pairs [Formula: see text] arising from root extensions are studied extensively. Transfer results for the “AV-ring” property are obtained for pullbacks of [Formula: see text] type, with applications to pseudo-valuation domains, integral minimal ring extensions, and integrally closed maximal non-AV subrings. Several sufficient conditions are given for [Formula: see text] being an AV-ring pair to entail that [Formula: see text] is an overring of [Formula: see text], but there exist domain-theoretic counter-examples to such a conclusion in general. If [Formula: see text] is an AV-ring pair and [Formula: see text] satisfies FCP, then each intermediate ring either contains or is contained in [Formula: see text]. While all AV-rings are quasi-local going-down rings, examples in positive characteristic show that an AV-domain need not be a divided domain or a universally going-down domain.