Quasi-quadratic modules in valuation rings and valued fields

Abstract

We define quasi-quadratic modules in a commutative ring generalizing the notion of quadratic modules. The main result is a structure theorem of quasi-quadratic modules in a subring A of a 2-henselian valued field $$(K,\text{val})$$ whose residue class field F is of characteristic $$\neq 2$$ . We further assume that the valuation ring B is contained in A. Set $$H=\rm{val}(A ^{\times} )$$ and $$G_{\geq e}=\{g \in G \mid g \geq e\}$$ . 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 $$\Theta\colon \mathfrak X_A \rightarrow \mathcal T_F^{ H \cup G_{\geq e}}$$ 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.