Let $A$ be a Unique factorization domain.
I want to prove :
Every non-invertible element of $A$ is divisible by at least one irreducible element.
My reason to believe that is because if $x$ is non invertible, either one can write $x=ab$ where $a$ is invertible (in that case $x$ is divisible by a smaller non-inverible $b$) , either one cannot write $x=ab$ and $x$ is itself irreducible.
There may be a recursion saying that, getting smaller and smaller non-invertibles, we end up with an irreducible.
Anyway. How to prove the statement ? Since I'm not fluent in algebra, I'd need a link with a detailed proof, including intermediate lemmas and definitions.