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I remind you that an Azumaya algebra $A$ is a central and separable algebra. Now, I know that if $A$ is an algebra over a skew-field or over a local ring then there exists a subalgebra $S$ of $A$ such that $S$ is commutative, maximal in $A$ and separable.

This result doesn't hold in general. Can you find a counterexample if the ring is not a local ring?

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