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 ($i\mathrm{.}e\mathrm{.}$, ${\chi }_0(x)=1\text{for all }x\in R^{\times }$), then the Gauss sum is $$G({\chi }_0,\psi )=\sum _{x\in R^{\times }}\psi (x)\mathrm{.}$$ 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.