Exercise 27, § 2

Let M be an A-module and Q a p-primary submodule of M. Q is said to be of exponent m if m is the smallest integer k such that $p^k M \subset Q$ (cf. Exercise 3); if there exists no integer k with this property, Q is said to be of infinite exponent. If Q is of finite exponent, Q is also said to be strongly primary.

(a) Show that, if $Q : p = Q$, Q is of infinite exponent and that there exists $\alpha \in p$ such that $Q : (\alpha)$ is distinct from Q and is a p-primary submodule of M of infinite exponent.

(b) Give an example of a ring A and a p-primary ideal q of infinite exponent and such that $q : p = q$ (cf. § 1, Exercise 2).

(c) Let K be a field, A the polynomial ring $K[X_n]_{n \in \mathbf{N}}$, p the prime ideal of A generated by the $X_n$ and q the p-primary ideal generated by the products $X_i X_j$ ($i \neq j$) and the elements $X_n^{n+1}$. Show that for all $a \in p - q$, $q : (a)$ is of finite exponent, that $q : p \neq q$ and that $q : p$ is a p-primary ideal of infinite exponent.

(d) Let Q be a p-primary submodule of M of finite exponent e. Show that the $p^k M$ for $k < e$ are all distinct and that the submodules $Q : p^k$ for $k < e$ are all distinct. For every ideal a of A show that either $Q : a = Q$ or $Q : a$ is a p-primary submodule in M distinct from M and of exponent $\leq e - 1$ (observe that if $a \subset p$ then $p^{e-1} M \subset Q : a$).