Models of Peano arithmetic as modules over initial segments (Q819263)

\textit{A. Tsuboi} and \textit{M. Murakami} proved in an unpublished preprint [see also their paper ``Expanding the additive reduct of a model of Peano arithmetic'', Math. Log. Q. 49, 363--368 (2003; Zbl 1026.03024)] that for every countable model \(M\) of PA there are a model \(N\) and its initial segment \(I\) such that the additive reducts of \(M\) and \(N\) coincide and, for all \(a\in M\), \(a\times^M a=a\times^N a\) iff \(a\in I\). The author proves a variant of this result which covers a large class of initial segments \(I\), but with a weaker conclusion. Let \(I\) be an initial segment of a model \(M\). If \(I\) is closed under \(+\) and \(\times\), then \(M\) can be considered in a natural way as an \(I\)-module over the field of fractions \(Q(I)\) (all models of arithmetic here are considered as rings). The main result is that if \(I\) is closed under \(+\), \(\times\), and the factorial function \(x\mapsto x!\), then there is a model \(N\), such that \(M\) and \(N\) coincide as \(I\)-modules, but \(\times^M\not=\times^N\).