Local ring with finite maximal ideal is finite

Let $I$ be a minimal right ideal; let $x\in I$, $x\ne 0$. What's the annihilator of $x$?

Let $R$ be not a field ; then $\exists 0 \ne x \in R\setminus U(R)$ . Then $Rx$ and $ ann(x)$ are both proper ideals of $R$ , hence both of them are finite . And obviously $Rx \cong R/ann(x)$ as $R$-modules ; hence $R/ann(x)$ is also finite . Thus $R$ is finite