Let $R$ be an integral domain. If $x \in R$ is prime, then $x$ is irreducible.

Question

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…

Answer 1

I think you were trying to make the argument that if $x =bc$, then one of them must be a unit (and you meant $x=bc$ in your first line.)

Then, it definitly follows that $x \mid bc$, and since this is a prime, then $x \mid b$ or $x \mid c$, and the rest works.