I am trying to understand the proof for the following theorem:
Let $R$ be an integral domain. If $x \in R$ is prime, then $x$ is irreducible.
Here is the proof:
I typed this a while ago and I don't understand the part where if $x | bc$, then $x=bc$? Is something wrong at this step?
Also, is the definition of prime elements where $p$ is prime if whenever $p|ab$, then either $p|a$ or $p|b$? Now I don't feel so sure. This could be the reason why I am not understanding the proof...