Gauss sum with principal multiplicative character

Let $R$ be a finite ring with unity, $\psi: R \to \mathbb{C}^\times$ be an additive character of $R$, and $\chi_0$ be the principal multiplicative character ($i.e.$, $\chi_0(x) = 1 \quad \text{for all } x \in R^\times$), then the Gauss sum is $$G(\chi_0, \psi) = \sum_{x \in R^\times} \psi(x).$$ In this paper, we give an explicit formula for a more general form of the Gauss sum $G(\chi_0, \psi)$. Interestingly, the formula extends the known formula of classical Ramanujan's sum to the context of finite rings. As an application, we derive the eigenvalues for a more general form of the unitary Cayley graph $\text{Cay}(R, R^{\times})$ using the formula.