Profinite Direct Sums with Applications to Profinite Groups of Type $\Phi_R$

We show that the "profinite direct sum" is a good notion of infinite direct sums for profinite modules having properties similar to direct sums of abstract modules. For example, the profinite direct sum of projective modules is projective, and there is a Mackey's Formula for profinite modules described using these sums. As an application, we prove that the class of profinite groups of type $\Phi_R$ is closed under subgroups.