In this paper, we investigate the Sombor index of the total graph and unit graph of $\mathbb{Z}_n$ which is denoted by $T_Γ(\mathbb{Z}_n)$ and $G(\mathbb{Z}_n)$ respectively for $n \in \{2k, p^α, pq, p^2q\}$ where $p$ and $q$ are distinct odd prime numbers such that $p < q$. Moreover, we compute the Sombor index of any finite local ring.