In commutative ring theory, defining irreducibility often takes several assimptions, for instance : $x$ is irreducible if
$x \neq 0$
$x$ is non-invertible
$x=ab \implies a$ invertible and $b$ non invertible or vice-versa
I think it's safe to say that an irreducible element can be defined as a non-invertible which isn't the product of several non-invertibles. But since I've nevet actually read this anywhere I'm a bit skeptical.
Is this definition equivalent to the three listed above ? If so, is it less convenient to state it that way ?