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. Further information: Primary decomposition theorem for ideals