= Simplicity of a Weyl algebra in characteristic zero
In a nonzero two-sided ideal choose an element of least <Bernstein filtration> degree. Commuting it with any position or differentiation generator lowers degree, so all such commutators vanish. The <ordered monomial basis of a Weyl algebra> and characteristic zero then force it to be a nonzero scalar. The ideal is the whole ring. The argument fails in positive characteristic, when powers of the generators can be central.
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