Exercise 23, § 3

(a) Let $A$ be a commutative Noetherian ring and $m$ an ideal of $A$. Let $P\{M\}$ denote the following property: $M$ is an $A$-module and every finitely generated submodule of $M$ is annihilated by a power of $m$.

Show that conditions (α), (β) and (γ) of Exercise 22 are fulfilled. (To prove (γ), reduce it to the case where $M$ is finitely generated; for all $a \in m$, there exists by hypothesis $k > 0$ such that $a^k N = 0$; use the fact that there exists $r > 0$ such that $\operatorname{Ker}(a_M^r) \cap \operatorname{Im}(a_M^r) = 0$ (Algebra, Chapter VIII, § 2, no. 2, Lemma 2).)

(b) Show that, if $M$ is a finitely generated $A$-module, the topology $\mathscr{T}_P(M)$ is identical with the $m$-adic topology. Give an example of an $A$-module $M$ for which $\mathscr{T}_P(M)$ is strictly finer that the $m$-adic topology (cf. Exercise 11).

(c) Show that the conclusion of (a) does not extend to the case where $A$ is a non-commutative left Noetherian ring and $m$ is a two-sided ideal of $A$ (consider the ring of lower triangular matrices of order 2 over a commutative field).