Exercise 3, § 2

Let A be a Noetherian ring and M a finitely generated A-module. Let Q be a p-primary submodule in M. The greatest lower bound of the integers $n \geq 1$ such that $p^n M \subset Q$ is called the exponent of Q in M and denoted by $e(M/Q)$.

Let $(\mathscr{Q}_\lambda)_{\lambda \in L}$ be a family of p-primary submodules in M. For $Q = \bigcap_{i} \mathscr{Q}_\lambda$ to be p-primary in M, it is necessary and sufficient that the family $(e(M/Q_\lambda))_{\lambda \in L}$ be bounded above; then $e(M/Q) \geq e(M/Q_\lambda)$ for all $\lambda \in L$.