Quasi-quadratic modules in pseudo-valuation domain

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 $(G,\le )$ 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 ${\prod }_{g\in G,g\ge e}{\mathfrak{X}}_{F_0}^F$.

20pages, 1 figure, Preprint submitted to a Journal