Exercise 6, § 4

a) Let A be a ring, M a faithful noetherian A-module. Show that A is a noetherian ring. b) Let A be a ring and M a faithful A-module of finite type. Assume that for every increasing sequence of ideals $(\alpha_i)_{i \in \mathbf{N}}$ of A, the sequence of submodules $(\alpha_iM)_{i \in \mathbf{N}}$ of M is stationary. Then A is a noetherian ring. (By a), it is enough to prove that M is a noetherian A-module. Reduce to the case where for every nonzero ideal $\alpha$ of A, the A-module $M/\alpha M$ is noetherian and where, for every nonzero submodule N of M, the A-module $M/N$ is not faithful.) c) Let B be a noetherian ring and A a subring of B such that B is a finite A-algebra. Then A is a noetherian ring. (This makes it possible to suppress the assumption that A be noetherian in Prop. 2 of § 4, No. 1.)