Structure of ultrapowers on Abelian groups over \(\omega\)
type: Item
instance of: scholarly article
title: Structure of ultrapowers on Abelian groups over \(\omega\)
zbMATH Open document ID: 0806.20045
DOI: 10.1007/BF02102117
author: A. M. Ivanov
published in: Mathematical Notes
publication date: 1994-02-08T00:00:00Z
review text: 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 \(G\).
Mathematics Subject Classification ID: 20K25
Mathematics Subject Classification ID: 03C20
zbMATH DE Number: 484237
zbMATH Keywords: algebraically compact group
zbMATH Keywords: system of invariants
zbMATH Keywords: \(\omega_ 1\)- saturated groups
zbMATH Keywords: ultrapowers of abelian groups
zbMATH Keywords: ranks
reviewed by: Grigore Călugăreanu
MaRDI profile type: Publication
cites work: Q5609457
description: scientific article; zbMATH DE number 484237