Exercise 15, § 1

Let A be a commutative ring and M a finitely generated A-module.

a) Suppose that for every increasing sequence $(\mathfrak{a}_n)_{n\in\mathbf{N}}$ of ideals of A, the sequence $(\mathfrak{a}_nM)_{n\in\mathbf{N}}$ is stationary. Prove that the A-module M is Noetherian. (Reduce to the case when the A-module M is faithful and when for every ideal $\mathfrak{a}\not= 0$ of A, the A-module $M/\mathfrak{a}M$ is Noetherian. There then exists a submodule $N_0$ of M that is maximal among the submodules $N\subset M$ such that $M/N$ is faithful. Prove that the A-module $M/N_0$ is Noetherian and then that the ring A is Noetherian, and conclude.)

b$)^*$Suppose that for every decreasing sequence $(\mathfrak{a}_n)_{n\in\mathbf{N}}$ of ideals of A, the sequence $(\mathfrak{a}_nM)_{n\in\mathbf{N}}$ is stationary. Prove that the A-module M is Artinian. (Reduce to the case when the A-module M is faithful. Prove that A then has only finitely many maximal ideals and, drawing on the proof of Proposition 1 of VIII, p. 173, that the radical of A is nilpotent. Deduce that there exists a finite sequence $(\mathfrak{m}_1,\mathfrak{m}_2,\cdots ,\mathfrak{m}_n)$ of maximal ideals of A whose product annihilates M. Conclude by induction on $n$.)\*