# Sh:831

- Göbel, R., & Shelah, S. (2005).
*How rigid are reduced products?*J. Pure Appl. Algebra,**202**(1-3), 230–258. DOI: 10.1016/j.jpaa.2005.02.002 MR: 2163410 -
Abstract:

For any cardinal \mu let {\mathbb Z}^\mu be the additive group of all integer-valued functions f:\mu\to {\mathbb Z}. The support of f is [f]=\{i\in\mu: f(i)=f_i\ne 0\}. Also let {\mathbb Z}_\mu= {\mathbb Z}^\mu/{\mathbb Z}^{<\mu} with {\mathbb Z}^{<\mu}= \{f\in {\mathbb Z}^\mu: \left|[f]\right|<\mu\}. If \mu\le \chi are regular cardinals we analyze the question when Hom({\mathbb Z}_\mu,{\mathbb 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({\mathbb 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. János Bolyai 61 (1992) 77–107) and a seminal result by Łoś 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 An interesting connection to earlier (unpublished) work on model theory by (unpublished, circulated notes, 1973) is elaborated in Section 3. - published version (29p)

Bib entry

@article{Sh:831, author = {G{\"o}bel, R{\"u}diger and Shelah, Saharon}, title = {{How rigid are reduced products?}}, journal = {J. Pure Appl. Algebra}, fjournal = {Journal of Pure and Applied Algebra}, volume = {202}, number = {1-3}, year = {2005}, pages = {230--258}, issn = {0022-4049}, mrnumber = {2163410}, mrclass = {20K15 (03E50)}, doi = {10.1016/j.jpaa.2005.02.002} }