Let $R$ be a P.I.D. and let $a \neq 0$ be an element in $R$. Prove that for a prime element $p$, $p(R/(a)) = ((p) + (a))/(a)$.
This is something that came up while reading Dummit and Foote's textbook page 466, Chapter 12.1 Lemma 8's proof. The author says that "note first that $p(R/(a))$ is the image of the ideal $(p)$ in the quotient $R / (a)$, hence is $(p) + (a)/ (a)$. I completely get that "$p(R/(a))$ is the image of the ideal $(p)$ in the quotient $R / (a)$" but I am not sure how it follows that $p(r/(a))$ "hence is $(p) + (a)/ (a)$". Shouldn't it be $(p)/(a)$ instead? I tried looking at the specific example of when $R = \mathbb Z$ and $p$ is a prime number, yet I am having difficulty seeing that it immediately follows the "hence is $(p) + (a)/ (a)$" part. Can you elaborate?