Need help understanding a step in a proof about modules over PIDs

Question

This is Chapter 3, Theorem 7.3 in Algebra by W. Adkins & S. Weintraub (GTM). It's about the uniqueness of the cyclic decomposition for finitely generated modules over a PID. I highlighted the exact step that I don't understand. How the negation of $\langle p \rangle \supseteq \operatorname{Ann}(z_i)$ implies that sum of ideals?

Answer 1

In a P.I.D., irreducible elements generate maximal ideals,. On the other hand, any proper ideal is contained in a maximal ideal. So, if $\operatorname{Ann}(z_i)$ is not contained in $(p)$, which is maximal, the only ideal which contains both is $R$.

Answer 2

In a PID, the ideal generated by a prime element is a maximal ideal.