Pure subgroups of \(A\)-projective groups
The following example will illustrate the topic considered by the authors. Let \(A =\bigoplus \mathbb{Z}_ p\) be the direct sum of the abelian groups \(\mathbb{Z}_ p\), the integers localized at the prime \(p\), where \(p\) runs over all primes. The endomorphism ring \(E(A) \cong \prod_ p \mathbb{Z}_ p\) of \(A\) is strongly non-singular and semi-hereditary. Recall that a ring \(E(A)\) is strongly non-singular if all finitely generated non-singular (right) \(E(A)\)-modules are the finitely generated submodules of free modules. Moreover \(A\) is flat over \(E(A)\) and satisfies the following two conditions. ``\(A\) splits pure \(A\)-socles'', i.e. all pure \(A\)-generated subgroups \(U\) (\(U = \sum\{\varphi(A): \varphi \in \text{Hom}(A,U)\})\) of \(A\)-projective groups \(D\) of finite rank (\(D\) is a summand of \(\bigoplus_ n A\)) are summands. ``\(A\) is not quasi-splitting \(A\)-socles'', i.e. there are subgroups \(U\) as above (but not pure) in \(D\) which are not even quasi-summands of \(D\). The paper gives a detailed analysis of the indicated properties for general torsion-free \(A\) and provides characterizations of them partly in terms of \(E(A)\)-modules (Theorem 2.1) or ring theoretically. Here is one of the results which can be applied to derive the properties of the above example. Corollary 2.3: If \(A\) is torsion-free abelian and faithful as \(E(A)\)-module, then the following are equivalent: (a) \(A\) splits pure \(A\)-socles and is a flat \(E(A)\)-module. (b) \(E(A)\) is (right) semi- hereditary and the quasi-endomorphism ring \(\mathbb{Q} E(A)\) is semi-simple artinian. A special case is Baer's splitting theorem for pure subgroups of homogeneous completely decomposable groups of finite rank.
homogeneous completely decomposable groups
scientific article; zbMATH DE number 638250