Exercise 32, § 2

A ring A is called primary if it is a local ring and its maximal ideal m is a nil ideal (*). Every ideal of A contained in m is then m-primary in A.

(a) Let A be a strongly Laskerian primary ring, so that in particular the maximal ideal m is nilpotent. Show that, if a, b are two ideals contained in m, necessarily $a : b \neq a$ (use Exercise 30 (b)).

(b) A primary Noetherian ring is Artinian. Deduce that, for every finitely generated module M over a Noetherian ring A and every p-primary submodule N of M, there exists an integer m such that every strictly decreasing sequence $(M_i/N)_0 \leq i \leq k$ of submodules of M/N for which the M_i are p-primary has length $k \leq m$ (consider the A,-module $M_p$ and note that $M_p/N_p$ is annihilated by a power of $pA_p$).

(c) Show that an Artinian ring A in which every prime ideal $\neq \{0\}$ is maximal is the direct composition of a finite number of primary rings.