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

Let $hat mathbb{Z}$ 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 → $hat mathbb{Z}$ linear mod H. (2) $hat mathbb{Z}$ is isomorphic to (ℤ₊/H )βH for some β : $hat mathbb{Z}$/H →ℚ linear mod H, though $hat mathbb{Z}$ is not the additive group of any model of Th(ℤ, +, ×) and the exact sequence H → $hat mathbb{Z}$ → $hat mathbb{Z}$/H is not splitting.