A generalization of Baer's splitting problem on Abelian groups
The paper treats a generalization of Baer's problem of describing all abelian groups A satisfying \(Ext(A,T)=0\) for all abelian torsion groups T. Here the given class of abelian groups is the full subcategory \({\mathcal C}_ G\) of the category \({\mathcal A}b\) of all abelian groups which is generated by an abelian group G. All abelian groups A with \(Ext(A,H)=0\) for all \(H\in {\mathcal C}_ G\) form another subcategory \({\mathcal A}_ G\) of \({\mathcal A}b\). The problem of characterizing the members of \({\mathcal A}_ G\) is reduced to torsion-free abelian groups. It turns out that a torsion- free abelian group A is in \({\mathcal A}_ G\) exactly if it is the union of a continuous well-ordered ascending chain of pure subgroups \(0=C_ 0<C_ 1<...<C_{\nu}<...<C_{\sigma}=A\) for some ordinal \(\sigma\) whose rank 1 factors \(B_{\nu}=C_{\nu +1}/C_{\nu}\) satisfy the conditions (i) \(pB_{\nu}\neq B_{\nu}\) whenever \(p^{k+1}G\neq p^ kG\) for all \(k\geq 0\) and all primes p; (ii) the type t of \(B_{\nu}\) can be realized by a characteristic \((...,k_ p,...)\) where \(k_ p=0\) for almost all primes p with pG\(\neq G\).
continuous well-ordered ascending chain of pure subgroups
scientific article; zbMATH DE number 4110949