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Baer’s lemma and Fuchs’ problem 84a. (English) Zbl 0592.20058

If A and G are abelian groups, then G is said to be A-projective if \(G\oplus H\cong \oplus_ IA\) for some index set I. The A-socle of G is defined as \(S_ A(G)=\sum \{f(A)|\) \(f\in Hom(A,G)\}\). We state the following conditions for further reference: (I) If U is a subgroup of an A-projective group with \(S_ A(U)=U\), then U is A-projective. (II) If B is a subgroup of G such that G/B is A-projective and \(S_ A(G)+B=G\), then B is a direct summand of G. (Ia) If B is a subgroup of \(\oplus_ IA\) with \(S_ A(B)=B\), then \(B\cong \oplus_ JA\). Finally, a group A is called self-small if Hom(A,-) preserves direct sums of copies of A.
The author generalizes Baer’s Lemma by proving the following result: Let A be a torsion-free, reduced, indecomposable abelian group. The following are equivalent: (a) A is self-small and flat as an E(A)-module, and E(A) is right hereditary. (b) A satisfies (I) and (II). Also, the following answer to Fuchs’ problem 84a is obtained: Let A be an abelian group. The following are equivalent: (a) E(A) is a principal ideal domain. (b) A belongs to one of the following classes of abelian groups: (\(\alpha)\) \(A\cong {\mathbb{Z}}(p)\) for some prime p of \({\mathbb{Z}}\). (\(\beta)\) \(A\cong {\mathbb{Z}}(p^{\infty})\) for prime p of \({\mathbb{Z}}\). (\(\gamma)\) \(A\cong J_ p\) for some prime p of \({\mathbb{Z}}\). (\(\delta)\) \(A\cong {\mathbb{Q}}\). (\(\epsilon)\) A is cotorsion-free (i.e. \({\mathbb{Z}}(p),{\mathbb{Q}},J_ p\subseteq A\) for all primes p of \({\mathbb{Z}})\) and (i) E(A) is commutative, (ii) A is indecomposable, and (iii) A satisfies conditions (Ia) and (II). Finally, some interesting applications of these results to locally A- projective groups are made. (G is locally A-projective if every finite subset of G is contained in a direct summand of G which is A-projective.)
Reviewer: P.Gräbe

MSC:

20K20 Torsion-free groups, infinite rank
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
16S50 Endomorphism rings; matrix rings
20K21 Mixed groups
20K99 Abelian groups
20K27 Subgroups of abelian groups
20K25 Direct sums, direct products, etc. for abelian groups
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