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In the following answer it is mentioned

Consider now a minimal element in the family of products of finitely many maximal ideals, let it be $M_1 \ldots M_k$.

Why should the family of products have a minimal element?

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Mohan
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At the beginning of the answer, it says

A finite dimensional (commutative or not) algebra over a field is an artinian ring, because it certainly satisfies the descending chain condition on left or right ideals.

The reason for this is that ideals are $K$-vector subspaces, and $\dim_KR$ is finite.

A ring is (left/right) artinian if and only if every non-empty family of (left/right) ideals has a minimal element.