Irreducible ideals are primary in Noetherian rings
= Irreducible ideals are primary in Noetherian rings
In a <Noetherian ring>, an <irreducible ideal> is a <primary ideal>. In its quotient, if $xy=0$ and $x\ne0$, choose $r$ after the <annihilators> of powers of $y$ stabilize. Then $(x)\cap(y^r)=0$, so irreducibility forces $y^r=0$. Combined with finite decomposition into irreducible ideals, this proves the <Lasker–Noether theorem>.