Let $A$ be a ring. A primary decomposition of an ideal $\mathfrak{a}\subseteq A$ is a collection of primary ideals $\{q_1,\dots ,q_k\}$ such that:

A primary decomposition is called minimal if:

1. For $i\ne j$, $r({\mathfrak{q}}_i)$ and ${\mathfrak{q}}_j$ are 1;;pairwise distinct, otherwise by the previous lemma, we can combine them
2. For each $i$, we have 1;;that $\bigcap _{j\ne i}{\mathfrak{q}}_i⊈{\mathfrak{q}}_j$, otherwise ${\mathfrak{q}}_i$ is superfluous.