Exercise 18, § 1

Let A be a ring and M an A-module. For an element $c \in A$ to be such that, for all $a \in A$ such that the homothety $a_M$ is injective, $b_M$ is an injective homothety for all $b$ of the form $a + \lambda c$, where $\lambda \in A$, it is sufficient that $c$ belong to the intersection of the maximal elements of $\operatorname{Ass}_f(M)$ (use Exercise 17 (b)). Is this condition necessary?