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 $R$.