Commutative Algebra
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Show that a finitely generated module over a Noetherian local ring is flat iff it is free using Nakayama and Tor.
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Show that ⟨2,x⟩ ⊴ is not a principal ideal.
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Let R be a Noetherian ring and A,B algebras over R. Suppose A is finite type over R and finite over B. Then B is finite type over R.