Topological aspects of irredundant intersections of ideals and valuation rings

TL;DR

This paper introduces a topological framework for analyzing irredundant intersections of ideals and valuation rings, focusing on spectral representations and their properties in various ring contexts.

Contribution

It develops a spectral space approach to irredundance, establishing topological criteria for minimal representations of intersections in algebraic structures.

Findings

Irredundance is characterized as a topological property in minimal spectral representations.

Application of the framework to valuation rings and ideals in specific classes of rings.

Identification of conditions under which representations are Noetherian and their implications.

Abstract

An intersection of sets $A = \bigcap_{i \in I}B_i$ is irredundant if no $B_i$ can be omitted from this intersection. We develop a topological approach to irredundance by introducing a notion of a spectral representation, a spectral space whose members are sets that intersect to a given set $A$ and whose topology encodes set membership. We define a notion of a minimal representation and show that for such representations, irredundance is a topological property. We apply this approach to intersections of valuation rings and ideals. In the former case we focus on Krull-like domains and Pr\"ufer $v$-multiplication domains, and in the latter on irreducible ideals in arithmetical rings. Some of our main applications are to those rings or ideals that can be represented with a Noetherian subspace of a spectral representation.