Primary decomposition theorem for ideals

Statement

Let $R$ be a Noetherian ring, and $I$ be a proper ideal in $R$. Then $I$ admits a primary decomposition, viz., there exists a finite collection $Q_{1},Q_{2},\ldots ,Q_{n}$ of primary ideals such that:

$I=\bigcap _{i=1}^{n}Q_{i}$

Further, the set of associated primes for $I$ (viewed as a $R$-module) is the same as the set of radicals for the $Q_{i}$s.

Also see primary decomposition theorem for modules.