Prove that $a$ is a prime element of $R$

Question

Let $R$ be a PID and $P = (a)$ is a prime ideal of $R$. Prove that $a$ is a prime element of $R$.

Since $P$ is a prime ideal of $R$, let $x,y \in R$ s.t. $xy \in P = (a).$ (WTS $a \mid x$ or $a\mid y$). Since, $xy \in P = (a) \Rightarrow x \in (a) \vee y \in (a) \Rightarrow a\mid x \vee a\mid y$. Hence, $a$ is a prime element of $R$.

Is this correct?

Answer 1

After making the prefix mentioned in the comments, you have a fine proof.

The reason why we need to change the assumption to "Let $x,y\in R$ such that $xy|a$" is that as currently worded, you slightly miss the actual question. You're only proving that $xy\in(a)\Rightarrow a|x\vee a|y$, whereas you need to prove that $a|xy\Rightarrow a|x\vee a|y$