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Differential operators in the language of modules

I am reading articles about differential operators. Authors try to treat differential systems , by studying the differential operator on a vector space in the language of modules. In this manner ...
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Subset of differential operators is a finitely generated module?

I was reading about differential operators, and there is a small claim I don't understand. First, let $A$ be a commutative algebra over $k$, a field. We have the recursive definition for the algebra ...