Primary decomposition

A primary decomposition expresses an ideal or submodule as an intersection of primary components.

Every proper ideal in a Noetherian ring is a finite intersection of primary ideals. This existence theorem does not give uniqueness of the individual components: embedded primary components may vary.

Over a commutative ring $R$, a proper submodule $N⊊M$ is primary when $rm\in N$ and $m\notin N$ imply $r^{k}M\subseteq N$ for some positive integer $k$. Equivalently, each scalar that kills a nonzero element of $M/N$ acts nilpotently on the entire quotient. Its radical annihilator is a prime ideal.

For example, $p^r\mathbb{Z}$ is a primary submodule of the integer module $\mathbb{Z}$ when $p$ is prime and $r\ge 1$: if a scalar kills a nonzero residue modulo $p^r$, it is divisible by $p$, and a sufficiently high power kills all residues. By contrast, $6\mathbb{Z}$ is not primary: $3\cdot 2\in 6\mathbb{Z}$, but $2\notin 6\mathbb{Z}$ and no power of three kills $\mathbb{Z}/6\mathbb{Z}$.