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How to prove that finitely generated ideals of finite algebras over a ring $F$ are finite over the $F$?

Looks like we can use Hilbert's basis theorem somehow?

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I would guess "finite" here means module-finite both for $A$ over $F$ and $I$ over $F$ (which means Deep's example fails) –  user73985 Jun 24 '14 at 14:19

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