On the homological significance of the group of rationals with square- free denominators (Q1099256)

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 \(\text{Ext}(R,A)\).