Let $R$ be a Noetherian ring and let $M$ be a finitely generated $R$-module.

(a) Show that every submodule of $M$ is finitely generated.

(b) Show that each maximal element of the set

$$\mathcal{A}=\{\operatorname{Ann}(m) \mid 0 \neq m \in M\}$$

is a prime ideal. [Here, maximal means maximal with respect to inclusion, and $\operatorname{Ann}(m)=\{r \in R \mid r m=0\} .]$

(c) Show that there is a chain of submodules

$$0=M_{0} \subseteq M_{1} \subseteq \cdots \subseteq M_{l}=M$$

such that for each $0<i \leqslant l$ the quotient $M_{i} / M_{i-1}$ is isomorphic to $R / P_{i}$ for some prime ideal $P_{i}$.