Fulton 2.9

2.47.* Suppose $R$ is a ring containing $k,$ and $R$ is finite dimensional over $k.$ Show that $R$ is isomorphic to a direct product of local rings.

Let $\{v_1,\ldots,v_n\}$ be a basis for $R$ over $k.$ Let $\varphi:k[X_1,\ldots,X_n]\to R$ be the natural map sending $X_i\mapsto v_i.$ Then $\varphi$ is a surjective ring homomorphism (and thus also a surjective vector space homomorphism over $k$ ). Let $I=\ker\phi.$ Note that $\dim_k(R)=\dim_k(k[X_1,\ldots,X_n]/I).$ By Lemma 2.14 $V(I)$ is finite. Then by Proposition 6, $R\cong k[X_1,\ldots,X_n]/I\cong \prod_{i=1}^N\mathcal{O}_i/I\mathcal{O}_i.$