Let $A$ be a local ring with maximal ideal $\mathfrak{m}$ and $B\supset A$ a ring which is finite as $A$-module over $A$.

(1) The map $f^\ast\colon \mathsf{Spec} \, B\longrightarrow \mathsf{Spec} \, A$ induced by the inclusion $A\subseteq B$ is surjective.

(2) $(f^\ast)^{-1}(\mathfrak{m})=\lbrace\mathfrak{m}_1,\ldots,\mathfrak{m}_n \rbrace$.

(3) If $A$ is $\mathfrak{m}$-adically complete, then $$B\simeq \prod_{i=1}^nB_{\mathfrak{m}_i}.$$

(4) If $A$ is $\mathfrak{m}$-adically complete find a primary decomposition of $\mathfrak{m}B$.

I'm able to solve the first three points of the exercise but the fourth. Can you help me with (4)?


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