Quasi-quadratic modules in pseudo-valuation domain

TL;DR

This paper investigates the structure of quasi-quadratic modules within pseudo-valuation domains, establishing a correspondence with modules over associated valuation rings and their residue fields.

Contribution

It introduces a classification of quasi-quadratic modules in pseudo-valuation domains via a correspondence with modules over valuation rings and residue fields.

Findings

Established a one-to-one correspondence between quasi-quadratic modules in the domain and modules over the residue field.

Extended the understanding of module structures in pseudo-valuation domains.

Connected the properties of modules in the domain with those in valuation rings and residue fields.

Abstract

We study quasi-quadratic modules in a pseudo-valuation domain $A$ whose strict units admit a square root. Let $\mathfrak X_R^N$ denote the set of quasi-quadratic modules in an $R$-module $N$, where $R$ is a commutative ring. It is known that there exists a unique overring $B$ of $A$ such that $B$ is a valuation ring with the valuation group $(G,\leq)$ and the maximal ideal of $B$ coincides with that of $A$. Let $F$ be the residue field of $B$. In the above setting, we found a one-to-one correspondence between $\mathfrak X_A^A$ and a subset of $\prod_{g \in G,g \geq e} \mathfrak X_{F_0}^F$.