Abelian groups with self-injective quasi-endomorphism rings.
This paper is a continuation of an extensive study by various authors of Abelian groups with respect to a fixed group \(A\). Typical concepts involved are `\(A\)-projective groups', namely the summands of direct sums of copies of \(A\), the `\(A\)-socle' \(\text{S}_A(G)=\Hom(A,G)A\) of a group \(G\), groups that are modules of the endomorphism ring \(E:=\text{E}(A)\) of \(A\). In order to obtain results various conditions must be imposed on \(A\). For example \(A\) is `fully faithful' as a left \(\text{E}(A)\)-module if for every non-zero right \(\text{E}(A)\)-module \(M\), it is true that \(M\otimes_EA\neq 0\), and \(A\) is `self-small' if for every map \(\alpha\in\Hom(A,\bigoplus_IA)\) there is a finite subset \(J\) of \(I\) such that \(\alpha(A)\subseteq\bigoplus_JA\). Of particular interest are groups \(A\) that have the `Baer splitting property', which means that for a group \(G\) with a subgroup \(B\) such that \(G/B\) is \(A\)-projective and \(B+\text{S}_A(G)=G\), it is true that \(G\cong B\oplus G/B\). The author considers these notions in the category of quasi-homomorphisms and thus has a ``quasi-Baer splitting property'', ``quasi-flat'' and ``quasi-faithful'' groups. There are numerous results. The proofs are homological in nature. One motivation for the paper is Fuchs' Problem~84 that asks for a characterization of the groups with self-injective endomorphism ring. It was shown by the author in an earlier paper [Commun. Algebra 15, 2451-2471 (1987; Zbl 0629.20027)] that torsion-free such groups must be divisible. The corresponding question in the quasi-homomorphism setting has more varied answers. As a sample we state Corollary~3.6. The following statements (a) and (b) are equivalent for a torsion-free group \(A\) of finite rank. (a) \(A\)-generated subgroups of \(A\) are \(\mathbb{Q} A\)-solvable and \(\mathbb{Q} E\) is self-injective. (b) \(A\) is quasi-flat and a subgroup \(B\) of a torsion-free group \(G\) of finite rank such that \(B/\text{R}_A(B)\) is \(\mathbb{Q} A\)-solvable is quasi-\(A\)-cobalanced if and only if \(\text{R}_A(B)=B\cap\text{R}_A(G)\).
scientific article; zbMATH DE number 2012021
scientific article