## 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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