In a noetherian ring $A$, every proper ideal $\mathfrak{b}\subseteq A$ has a primary decomposition.

PROOF:

1. Say an ideal $\mathfrak{b}\subseteq A$ is irreducible if $\mathfrak{b}=\mathfrak{a}\bigcap \mathfrak{c}$ implies 1;; $\mathfrak{b}=\mathfrak{a}$ or $\mathfrak{b}=\mathfrak{c}$.
2. Show that every ideal is a intersection of finitely many irreducible ideals. To do this, assume contradiction and let $\mathrm{\Sigma }=$1;;$\{\mathfrak{a}\subseteq A|\mathfrak{a}\text{ is not an intersection of finitely many irreducible ideals}$
3. Show that every irreducible ideal is primary. (Here we may assume $\mathfrak{a}=0$, and we look for a proof with annihilators).