Minimal primary decomposition

A minimal primary decomposition of an ideal $I$ is an irredundant finite intersection $I={\bigcap }_i{Q}_i$ of primary ideals whose radicals are pairwise distinct.

An isolated prime of a minimal primary decomposition is a minimal member of the set of radicals of its primary components.

In a Noetherian ring, the primary component belonging to each isolated prime $\mathfrak{p}$ is uniquely determined byEmbedded primary components need not be unique.