Let $A$ be a commutative algebra over the finite field $\mathbb F_q$, of order $q$, where $q=p^l$, for a prime $p$ and a positive inteher $l$. Assume that $A$ is finite-dimensional. So, $A$ is a finite ring. Hence, we have the decomposition
where each $R_i$ is alocal ring. What can be said about the residue fields of ecah local ring $R_i$ ? Are all equal to $\mathbb F_q$?