Narkewicz introduced a class of generalized arithmetical convolutions [Nar63]. Burnett and Osterman introduced the concept of an A-function to unify the treatment of these convolutions with generalized divisibility relations and generalizations of multiplicativity of arithmetical functions [BO19]. They also considered some sets of arithmetical functions that form groups under special convolutions. We generalize these results and prove necessary and sufficient conditions for these sets of arithmetical functions to form groups and rings under certain convolutions. Specifically, we introduce a special class of A-functions, which we call perfect, and prove that they correspond to rings of arithmetical functions with respect to the A-convolution and standard pointwise function multiplication.

Finally, we introduce analogues of several of our results in the context of Davison K-convolutions that help illuminate possible future work.

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