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.