On the Additive Group Structure of the Nonstandard Models of the Theory of Integers
Makaleler > Tam Makale · 2002
Yazarlar: Hasan Dalgin, Labib Haddad, Mehmet Terziler
MLQ, Uluslararası, Wos, 2002, cilt 48, ss 403-412
DOI: 10.1002/1521-3870(200204)48:3<403::aid-malq403>3.0.co;2-g
Özet
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 +βis 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.