Let equation image 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 , where +βis defined by +β = + β — β) for a certain β : F → G linear mod H meaning that β = 0 and β + β — β ∈ H for all a, b in F. In this paper we show that the following hold: The (...) additive group of any nonstandard model ℤ* of the ring ℤ is isomorphic to βH for a certain β : ℤ*+/H → equation image linear mod H. equation image is isomorphic to βH for some β : equation image/H →ℚ linear mod H, though equation image is not the additive group of any model of Th and the exact sequence H → equation image → equation image/H is not splitting. (shrink)