Exercise 32, § 2

An ring $A$ is said to be primary if it is a local ring and if its maximal ideal $m$ is a nilideal (*). 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, one necessarily has a : b ≠ a (use exerc. 30 b)).

(*) For artinian (commutative) rings, this definition coincides with that of Alg., chap. VIII, § 6, exerc. 20.

b) A noetherian primary ring is artinian. Deduce that for every module M of finite type over a noetherian ring Λ 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-primaries, has a length k ≤ m (consider the A_p-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 ≠ {0} is maximal is a direct product of a finite number of primary rings.