One step of the proof is the following.
Let $R$ be a PID. Let $x$ be a non-zero, non-unit element in $R$.
Suppose $x$ can not be written as a product of irreducible elements, then $x$ is not irreducible.
I tried to make a proof for this proposition, using contraposition, i tried to find a proof for: if $x$ is irreducible, then $x$ can be written as a product of irreducible elements. When I started the proof, I found something that in my mind means that the proposition is false. However, I know that my own proof must be wrong, but I can't spot the mistake.
So assume $x$ is irreducible. If $x = a \cdot b $ then $a$ or $b$ has to be a unit. And surely a unit is not irreducible. So we can never write $x$ as a product of irreducible elements.
So the main question is: why does my attempt not hold and how do you go about making a valid proof.
I'm new to the subject and I understand that I could be making 'dumb' mistakes.
Thank you in advance.
Best Answer
If $x$ is irreducible then it is the product of one irreducible element, $x$. Products need not be of two or more elements. You can even take an empty product, which is defined to be $1$.