Exercise 30, § 2

Let M be an A-module and let N be a submodule of M admitting a reduced primary decomposition N = \bigcap_{i=1}^{n} Q_i; let p_i be the prime ideal associated with Q_i.

(a) Show that, if b is an ideal of A which is not contained in any of the p_i, then N : b = N (cf. Exercise 17).

(b) If each of the Q_i is strongly primary in M, show conversely that, if N : b = N, b is not contained in any of the p_i. (Note that, if b \subset p_i and P = \bigcap_{j \neq i} Q_j, then b^rP \subset N for some suitable integer r and deduce that, if N : b = N, then P = N.) Deduce that there then exists \beta \in b such that N : (\beta) = N.