Quasi-quadratic modules in valuation ring and valued field

Abstract

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 $(K,\mathbf{val})$ whose residue class field $F$ of characteristic $\ne 2$. We further assume that the valuation ring $B$ is contained in $A$. Set $H=\mathbf{val}(A^{\times })$ and $G_{\ge e}=\{g\in G|g\ge e\}$. 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 ${\mathcal{T}}_F^{H\cup G_{\ge e}}$ of ${\prod }_{g\in H\cup G_{\ge e}}{\mathfrak{X}}_F$. We explicitly construct the map $\Theta :{\mathfrak{X}}_A\to {\mathcal{T}}_F^{H\cup G_{\ge e}}$ 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.

41pages, 1 figure, Preprint submitted to a Journal

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