Exercise 22, § 2

Let E be an A-module and F a submodule of E. (a) Let p be a prime ideal of A and $a$ an element of A such that $a \notin p$ and $Q = F : (a)$ is p-primary in E. Show that $F = Q \cap (F + aE)$. (b) Suppose that there exist $b \in A$ and $x \in E$ such that $b^n x \notin F$ for all $n > 0$. Show that, if there exists an integer $n > 0$ such that $F : b^{n+1} = F : b^n$, then $F = (F + Ab^n x) \cap (F : (b^n))$. (c) Suppose that F is irreducible with respect to E (Algebra, Chapter II, § 2, Exercise 16) and consider the three following properties: $(\alpha)$ F is primary in E. $(\beta)$ For every prime ideal p of A, the saturation of F with respect to p in E is of the form $F : (a)$ for some $a \notin p$. $(\gamma)$ For all $b \in A$, the sequence $(F : (b^n))_{n \geq 1}$ is stationary. Show that each of conditions $(\beta)$, $(\gamma)$ implies $(a)$ (use (a) and (b) and Exercise 18 (a)). If E is finitely generated, the three conditions $(a)$, $(\beta)$ and $(\gamma)$ are equivalent (cf. Exercise 20 (b) and 20 (c)). (d) Let K be a field, A the ring $K[X, Y]$ and m the maximal ideal $AX + AY$ of A; show that the ideal $m^2$ is primary but not irreducible. (e) Let A be a not necessarily commutative ring and E a left Noetherian A-module. Show that every submodule F of E is the intersection of a finite family of irreducible submodules in E (consider a submodule of E which is maximal among those which are not finite intersections of irreducible submodules). (f) Deduce from (e) and (c) a new proof of Theorem 1 of no. 2.