Z-kernel groups of measurable cardinalities (Q1086360)

Z-kernel (abelian) groups are obtained by closing Z with respect to forming direct sums and direct products over sets of indices of arbitrary cardinality (not necessarily non-measurable). A known notion of type (being a symbol (\(\mu\),P), (\(\mu\),S), (\(\mu\),M)) of such a group [see \textit{M. Dugas}, \textit{B. Zimmermann-Huisgen}, Lect. Notes Math. 874, 179- 193 (1981; Zbl 0475.13011)] is shown to be well defined and unique in this more general case for Z-kernel groups in a fixed finitely definable universe \({\mathcal M}\), under the assumption that \(\mu\) is smaller than the least measurable cardinal \(M_ c\). It is proved that 1. If G, G' are Z- kernel groups in \({\mathcal M}\) of types (\(\mu\),X), (\(\mu\) ',X') and \(\mu <M_ c\) (or \(\mu '<M_ c)\) and G is isomorphic to a direct summand of G', then either \(\mu <\mu '\) or \(\mu =\mu '\) and \(X'=M\). 2. If G is as above then Hom(G,Z) is a Z-kernel group and its type is (\(\mu\),P), (\(\mu\),S) or (\(\mu\),M) according to \(X=S,P,M\).