On page 52 they write "...By (4.3) we can achieve (i)..."
where (4.3) is the lemma on the previous page that states that if $q_i$ are all $p$-primary then $\bigcap_i q_i$ is $p$-primary and (i) is the property of a minimal primary decomposition that $r(q_i)$ are pairwise distinct.
I'm confused about how $r(q_i)=p$ for all $i$ helps us to get $r(q_i) \neq r(q_j)$ for all $i \neq j$. If $r(q_i)=p$ for all $i$ then the primary decomposition would only consist of one ideal because we'd have to throw all the others away to get (i). Or not?