Structure of ultrapowers on Abelian groups over \(\omega\)
It is known that if \(G\) is an algebraically compact group then the invariants of its maximal divisible subgroup along with the invariants of its \(p\)-adic components constitute a complete, independent system of invariants for \(G\). Ultraproducts \(E\) are \(\omega_ 1\)-saturated groups [\textit{Yu. Ershov}, Decision problems and constructivizable models (1980; Zbl 0495.03009)] and hence algebraically compact groups. Thus, in order to determine the structure of the ultrapowers of abelian groups over \(\omega\) the author computes these invariants for \(A^ \omega\). The following ranks are characterized for a group \(G\) in order to be isomorphic to some ultrapower \(A^ \omega: r(p^ k G[p]/p^{k + 1} G[p])\), \(r_ 0(p\)-basic subgroup of \(G\)), \(r_ p(D(G))\) and \(r_ 0(D(G))\) where \(D(G)\) is the maximal divisible subgroup of \(G\).
scientific article; zbMATH DE number 484237