The group ring of $\mathbb{Q}/\mathbb{Z}$ and an application of a divisor problem

First we prove some elementary but useful identities in the group ring of $\mathbb{Q}/\mathbb{Z}$. Our identities have potential applications to several unsolved problems which involve sums of Farey fractions. In this paper we use these identities, together with some analytic number theory and results about divisors in short intervals, to estimate the cardinality of a class of sets of fundamental interest.