A Dedekind-Mertens theorem for power series rings

TL;DR

This paper extends the Dedekind-Mertens lemma to power series rings, providing new theoretical insights, counterexamples, and applications to integrality, while also correcting previous errors in the literature.

Contribution

It introduces a power series ring analogue of the Dedekind-Mertens lemma, advancing the understanding of algebraic properties in this context.

Findings

Established a Dedekind-Mertens theorem for power series rings

Provided counterexamples illustrating limitations or boundaries

Applied the theorem to problems of integrality

Abstract

We prove a power series ring analogue of the Dedekind-Mertens lemma. Along the way, we give limiting counterexamples, we note an application to integrality, and we correct an error in the literature.