On the additive group structure of the nonstandard models of the theory of integers

Let (Z) over cap (+) denote the inverse limit of all finite cyclic groups. Let F, G and H be abelian groups with H less than or equal to G. Let FbetaH denote the abelian group (F x H, +(beta)), where +(beta) is defined by (alpha, x) +(beta) (b, y) = (a + b, x + y + beta(a) + beta(b) - beta(a + b)) for a certain beta : F --> G linear mod H meaning that beta(0) = 0 and beta(a) + beta(b) - beta(a + b) epsilon 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)betaH for a certain beta:Z(+)*/H --> (Z) over cap (+) linear mod H. (2) (Z) over cap (+) is isomorphic to ((Z) over cap (+)/H)betaH for some beta:(Z) over cap (+)/H --> Q linear mod H, though (Z) over cap (+) is not the additive group of any model of Th(Z,+, x) and the exact sequence H --> (Z) over cap (+) --> (Z) over cap (+)/H is not splitting.