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Cardinality restrictions on preradicals. (English) Zbl 0687.20049

Abelian group theory, Proc. 4th Conf., Perth/Aust. 1987, Contemp. Math. 87, 277-283 (1989).
[For the entire collection see Zbl 0666.00004.]
A preradical T is a subfunctor of the identity for abelian groups. Preradicals are the basis for a general theory of socles and radicals in module theory. In the first case \(T(A/TA)=0\) for all objects A. The paper is devoted to a detailed discussion of preradicals related with cardinals and axioms of set theory.
If \(\kappa\) is cardinal, we can derive from T a new preradical \(T^{[\kappa]}\) defined by \(T^{[\kappa]}A=\sum \{TX:\) \(X\subseteq A\), X is generated by \(<\kappa\) elements\(\}\). A radical satisfies c.c. (cardinal condition) provided there exists some \(\kappa\) such that \(T=T^{[\kappa]}\). Recent progress on abelian groups made it possible to settle questions on c.c. in module theory. The paper under discussion follows this line. If \(R_{{\mathbb{Z}}}\) denotes the well-known Chase- radical \((R_{{\mathbb{Z}}}A=\cap \{\ker h:\) \(h\in Hom(A,{\mathbb{Z}})\})\), then it is shown among other things that \(R_{{\mathbb{Z}}}^{[\aleph_ 1]}\neq R_{{\mathbb{Z}}}^{[\aleph_ 2]}\), which gives more inside into complicated torsion-free abelian groups. The given examples are non- trivial and interesting for readers familiar with torsion-free abelian groups or those interested in radical theory.
Reviewer: R.Göbel

MSC:

20K27 Subgroups of abelian groups
16Nxx Radicals and radical properties of associative rings
20K20 Torsion-free groups, infinite rank
03E55 Large cardinals

Citations:

Zbl 0666.00004