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If a ring A is a subring of a commutative Noetherian ring B such that B is a finitely generated module over A, then A is a Noetherian ring.
*Similarly, if a ring A is a subring of a commutative Noetherian ring B such that B is faithfully flat over A (or more generally exhibits A as a pure subring), then A is a Noetherian ring (see the "faithfully flat" article

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