On the elementary theory of rings of Witt vectors with a multiplicative set of representatives for the residue field
Let \(T\) be a theory in the language of rings with extra unary predicate \(S\) whose models are \((B,\Sigma)\) (\(\Sigma\) corresponds to \(S\)) such that (1) \(B\) is a valuation ring with fraction field \(F\) of characteristic 0. (2) \(\Sigma\) is a multiplicative closed subset of \(B\) that is mapped bijectively onto \(B/m(B)\) by the residue class map. (3) \(m(B)= pB\) and \(B/m(B)\) is a perfect field. (4) The local ring \(B\) is Henselian. (5) For each \(k\in\mathbb{Z}^n\), \(k\cdot\sigma= 0\Leftrightarrow R(\overline\sigma)= 0\), for all \(\sigma\in \Sigma^n\), where \(R\in \mathbb{F}_p[X]^N\) is the triple associated to \(k\), and \(\overline\sigma= (\overline\sigma_1,\dots,\overline\sigma_m)\). The author proves that two models \((B,\Sigma)\) and \((B',\Sigma')\) of \(T\) are elementarily equivalent if and only if \(B/m(B)\) and \(B'/m(B')\) are elementarily equivalent and their value groups are elementarily equivalent.
scientific article; zbMATH DE number 1275768