Primary decomposition of an ideal

Definition
Let $$R$$ be a commutative unital ring, and $$I$$ a proper ideal in $$R$$. A primary decomposition of $$I$$ is an expression of $$I$$ as an intersection of finitely many primary ideals.

When $$R$$ is a Noetherian ring, every proper ideal admits a primary decomposition, and this primary decomposition has certain uniqueness properties.