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 \(\left\langle{ 2, x }\right\rangle{~\trianglelefteq~}{\mathbf{Z}}[x]\) 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 \(R\).