Baer’s lemma and Fuchs’s problem 84a

Abstract:

An indecomposable, torsion-free, reduced abelian group $A$ has the properties that (i) each subgroup $B$ of an $A$-projective group with ${S_A}(B) = B$ is $A$-projective and (ii) each subgroup $B$ of a group $G$ with ${S_A}(G) + B = G$ and $G/B$ $A$-projective is a direct summand if and only if $A$ is self-small and flat as a left $E(A)$-module, and $E(A)$ is right hereditary. Furthermore, a group-theoretic characterization is given for torsion-free, reduced abelian groups with a right and left Noetherian, hereditary endomorphism ring. This is applied to Fuchs’ Problem 84a. Finally, various applications of the results of this paper are given.