I have to show the following:
Let $p \in R\setminus\{0\}$ then: $p$ is prime element in $R$ if and only if $(p)$ is a prime ideal in $R$.
I have real problems doing so. I tried the following:
$\Rightarrow$ let $p$ be a prime element in $R$, then we know that if $p \mid ab$ then $p\mid a$ or $p\mid b$. Also, we know that $(p) = Rp = \{ rp \mid r \in R\},$ then we know that $(p)$ is prime ideal because $rp \in (p) \Rightarrow p \in (p)$.
Now how do i show that $(p)\neq R$. Also, the last step feels strange, as this seems to imply that $(a)$ is prime ideal for any $a\in R$ if $(a) \neq R$?
with the $\Leftarrow$ direction I do not know how to start, could I get any hints?
Thanks!