A generalization of Ramanujan's sum over finite rings

Abstract: Let $R$ be a finite ring with unity. In general, the eigenvalues of the unitary Cayley graph $\text{Cay}(R, R^{\times})$ are not known when $R$ is a non-commutative. In this paper, we present an explicit formula for the eigenvalues of $\text{Cay}(R, R^{\times})$ for any finite ring $R$. However, our focus is on a more general case of the unitary Cayley graph. It is well known that the classical Ramanujan's sum represents the eigenvalues of $\text{Cay}(\mathbb{Z}_n, \mathbb{Z}_n^{\times})$. Consequently, the eigenvalues of $\text{Cay}(R, R^{\times})$ can be view as a generalization of classical Ramanujan's sum in the context of finite rings. Interestingly, the formula we derive for the eigenvalues of $\text{Cay}(R, R^{\times})$ extends the known formula of classical Ramanujan's sum to the context of finite rings.