A generalization of Ramanujan's sum over finite rings

Let R be a finite ring with unity. In general, the eigenvalues of the unitary Cayley graph Cay(R, R×) are not known when R is a non-commutative. In this paper, we present an explicit formula for the eigenvalues of Cay(R, R×) 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 Cay(Zₙ, Zₙ×). Consequently, the eigenvalues of Cay(R, R×) 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 Cay(R, R×) extends the known formula of classical Ramanujan's sum to the context of finite rings.