On the homological significance of the group of rationals with square- free denominators
Frattini subgroups and Frattini closures of Abelian groups were studied by L. Fuchs and V. Dlab as early as in 1960. Now it turns out that for torsion-free Abelian groups these constructions are obtained quite easily by applying the functors \(\text{Hom}(R,A)\) and \(R \otimes A\), respectively, where \(R\) denotes the group of rationals with square-free denominators. Furthermore, the group \(R\) is closely related to the natural diagonal homomorphism \(\Delta_A : A \to \prod_{(p\text{ prime})}A / pA\), for \(A\) Abelian. This homomorphism is epi iff \(\text{Ext}(R,A) = 0\). In particular, \(\text{Ext}(R,A) = 0\), if the set \(\Sigma(A)\) of primes \(p\) with \(A/pA \neq 0\) is finite. Conversely, it is shown that for Frattini closures A finiteness of \(\Sigma(A)\) is equivalent to the vanishing of \(\text{Ext}(R,A)\).
scientific article; zbMATH DE number 4040180