Quasi-quadratic modules in valuation ring and valued field

This is a revised version of the previous version with a new appendix consisting of characteristic two case. We define quasi-quadratic modules in a commutative ring generalizing the notion of quadratic modules. The main theorem is a structure theorem of quasi-quadratic modules in a subring $A$ of a $2$-henselian valued field $(K,{\bf val})$ whose residue class field $F$ of characteristic $\neq 2$. We further assume that the valuation ring $B$ is contained in $A$. Set $H={\bf val}(A^\times)$ and $G_{\geq e}=\{g \in G\;|\; g \geq e\}$. The notation $\mathfrak X_R$ denotes the set of all the quasi-quadratic modules in a commutative ring $R$. Our structure theorem asserts that there exists a one-to-one correspondence between $\mathfrak X_A$ and a subset $\mathcal T_F^{ H \cup G_{\geq e}}$ of $\prod_{g \in H \cup G_{\geq e}}\mathfrak X_F$. We explicitly construct the map $\Theta: \mathfrak X_A \rightarrow \mathcal T_F^{ H \cup G_{\geq e}}$ and its inverse. We also give explicit expressions of $\Theta(\mathcal M \cap \mathcal N)$ and $\Theta(\mathcal M+\mathcal N)$ for $\mathcal M, \mathcal N \in \mathfrak X_A$. In addition, we briefly investigate the case in which the field $F$ is of characteristic two in the appendix as well.