Relatively diophantine correct models of arithmetic (Q1097277)

If M and N are models of Peano Arithmetic and M is an extension of N, then M is called N-diophantine correct of every polynomial with coefficients from M which has a root in M has already a root in N. It is shown that, for countable N and M, M is N-diophantine correct iff for every a in M, \(N<a\), there is an embedding f of M onto an initial segment I of M such that \(I<a\) and f is the identity function on N. This generalizes a result of \textit{L. Lipshitz} [Proc. Am. Math. Soc. 73, 107- 108 (1979; Zbl 0402.03055)] where the same is proven in the case when N is the standard model. The proof is based on the Matijasevič theorem and a back-and-forth construction.