Let $R$ be a noetherian integral domain. I want to show that any non-zero and non-invertible element $a$ can be written as a finite product of irreducible elements.
my ideas: I should argue by contradiction and consider the set $M$ of the ideals generated by elements, which cannot be written as a product of irreducibles. Since R noetherian, we find a maximal element $(b)$ of $M$. Moreover we find a minimal overlying prime ideal $I$ of $(b)$. But now I'm stuck. I wanted to show that all minimal overlying prime ideals of $(b)$ are principal ideals, but that doesn't hold generally (I think).