Quasi-quadratic modules in pseudo-valuation domain

We study quasi-quadratic modules in a pseudo-valuation domain A whose strict units admit a square root. Let \(\mathfrak X_R^N\) ${\mathfrak{X}}_R^N$ 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 \((G,\le )\) $(G,\le )$ 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 \({\mathfrak {X}}_A^A\) ${\mathfrak{X}}_A^A$ and a subset of \(\prod _{g \in G,g \ge e} {\mathfrak {X}}_{F_0}^F\) ${\prod }_{g\in G,g\ge e}{\mathfrak{X}}_{F_0}^F$ .