I'm looking for help with problem #6 on page 189 of Jacobson's "Basic Algebra I". In particular, we suppose $D$ is a PID and $M$ is a finitely generated module over $D$ with generators $x_1, ... x_n$. Furthermore, we specify these generators have minimal $n$ and that $l(x_1)$ is minimal among generating sets with $n$ members. ($l(x)$ is defined to be number of primes in the factorization of $a \in D$ where $\text{ann}(x) = (a)$. If $a=0$ then we define $l(a) = \infty$.)
Define $N := \sum_{j \ge 2}{Dx_j}$. The problem claims "To show $\text{ann}(x_1) \supset \text{ann}(y)$ for $y \in N$ it suffices to prove $\text{ann}(x_1) \supset \text{ann}(x_j), j \ge 2$". Can someone clarify for me why this is sufficient?