A book I am reading defines irreducible elements of an integral domain $D$ as follows:
A nonzero element $a\in D$ is called an irreducible if $a$ is not a unit and, whenever $b$, $c \in D$ with $a=bc$, then $b$ or $c$ is a unit.
In a proof following this definition, the author writes:
If $a$ is irreducible, we are done. Thus, we may assume that $a = bc$, where neither $b$ nor $c$ is a unit and $c$ is nonzero.
My question is:
Is a factorisation of an element which is not irreducible (here, $a$ with $a=bc$) always guaranteed?
Obviously, if such a factorisation exists, then neither $b$ nor $c$ can be units (else this would contradict the non-irreducibility of $a$).
I have read this post: Reducible element of ring can be factorized, however it hasn't helped.
Help!