How rigid are reduced products?

by Goebel and Shelah. [GbSh:831]
J Pure and Applied Algebra, 2005
For any cardinal mu let Z^mu be the additive group of all integer-valued functions f: mu-> Z . The support of f is [f]= {i in mu : f(i)=f_i ne 0} . Also let Z_mu = Z^mu / Z^{< mu} with Z^{< mu}= {f in Z^mu : |[f]|< mu} . If mu <= chi are regular cardinals we analyze the question when Hom (Z_mu, Z_chi) = 0 and obtain a complete answer under GCH and independence results in Section 8. These results and some extensions are applied to a problem on groups: Let the norm |G | of a group G be the smallest cardinal mu with Hom (Z_mu,G) ne 0 - this is an infinite, regular cardinal (or infty). As a consequence we characterize those cardinals which appear as norms of groups. This allows us to analyze another problem on radicals: The norm |R | of a radical R is the smallest cardinal mu for which there is a family {G_i: i in mu} of groups such that R does not commute with the product prod_{i in mu}G_i . Again these norms are infinite, regular cardinals and we show which cardinals appear as norms of radicals. The results extend earlier work (Arch. Math. 71 (1998) 341--348; Pacific J. Math. 118(1985) 79--104; Colloq. Math. Soc. J{a}nos Bolyai 61 (1992) 77--107) and a seminal result by {L}o{s} on slender groups (His elegant proof appears here in new light; Proposition 4.5.), see Fuchs [Vol. 2] (Infinite Abelian Groups, vols. I and II, Academic Pess, New York, 1970 and 1973). An interesting connection to earlier (unpublished) work on model theory by (unpublished, circulated notes, 1973) is elaborated in Section 3.

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