The quasi-Baer-splitting property for mixed Abelian groups.
Quasi decomposition theory for Abelian groups has been of great interest over the last decades. Hence the Abelian category \(\mathbb{Q} Ab\) with all Abelian groups as objects and morphisms \(\mathbb{Q}\otimes\Hom(A,C)\) as morphisms between Abelian groups \(A\) and \(C\) has been considered. Many concepts in the theory of Abelian groups have a quasi-analogue, i.e. a similar concept in the quasi category. The author of this paper studies the quasi Baer splitting property. An Abelian group \(C\) is called `almost \(A\)-generated' for some Abelian group \(A\) if there exists an epimorphism \(A^{(I)}\to C\) in \(\mathbb{Q} Ab\). Moreover, \(C\) is called `almost \(A\)-projective' if it is isomorphic to a quasi-summand of a direct sum of copies of \(A\). Accordingly, the Abelian group \(A\) has the `quasi Baer splitting property' if every epimorphism \(C\to G\to 0\), with \(C\) an almost \(A\)-generated group and \(G\) an almost \(A\)-projective group, quasi-splits. The author proves various characterizations of self-small groups which have the quasi Baer splitting property. Further results are shown, e.g. a self-small group \(A\) is almost \(E\)-flat and has the quasi Baer splitting property if and only if the class of almost \(A\)-adstatic modules is closed with respect to submodules and if and only if \(A\) is almost-projective with respect to the class of almost \(A\)-static groups.
scientific article; zbMATH DE number 2093439
scientific article