On the Additive Group Structure of the Nonstandard Models of the Theory of Integers

Let \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\hat \mathbb{Z}$\end{document} denote the inverse limit of all finite cyclic groups. Let F , G and H be abelian groups with H ≤ G . Let FβH denote the abelian group ( F × H , + β ), where + β is defined by ( a , x ) + β ( b , y ) = ( a + b , x + y + β ( a ) + β ( b ) — β ( a + b )) for a certain β : F → G linear mod H meaning that β (0) = 0 and β ( a ) + β ( b ) — β ( a + b ) ∈ H for all a , b in F . In this paper we show that the following hold: (1) The additive group of any nonstandard model ℤ* of the ring ℤ is isomorphic to (ℤ* + / H ) βH for a certain β : ℤ* + / H → \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\hat \mathbb{Z}$\end{document} linear mod H . (2) \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\hat \mathbb{Z}$\end{document} is isomorphic to (ℤ + / H ) βH for some β : \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\hat \mathbb{Z}$\end{document} / H →ℚ linear mod H , though \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\hat \mathbb{Z}$\end{document} is not the additive group of any model of Th(ℤ, +, ×) and the exact sequence H → \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\hat \mathbb{Z}$\end{document} → \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\hat \mathbb{Z}$\end{document} / H is not splitting.