Exercise 29, § 2

Let $A$ be a commutative ring, $\mathfrak{m}$ an ideal of $A$ such that $\bigcap_{n \geq 0} \mathfrak{m}^n = 0$, $c$ an element of $A$ which is not a divisor of 0. Now prove that the transporters $q_n = \mathfrak{m}^n : Ac$ are open for the $\mathfrak{m}$-adic topology and have an intersection reduced to 0. If $A$ is strictly linearly compact for the $\mathfrak{m}$-adic topology (Exercise 22 b)), $(q_n)$ is a fundamental system of neighborhoods of 0 in $A$ for this topology.