909 Extra Problems Commutative Algebra

Commutative Algebra

  • Show that a finitely generated module over a Noetherian local ring is flat iff it is free using Nakayama and Tor.

  • Show that \(\left\langle{ 2, x }\right\rangle{~\trianglelefteq~}{\mathbf{Z}}[x]\) is not a principal ideal.

  • 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 \(R\).