Primary decomposition

Summary

A primary decomposition of an ideal writes it as a finite intersection of primary ideals. It generalises the factorisation of an integer into prime powers, since $(n) = (p_1^{e_1}) \cap \cdots \cap (p_k^{e_k})$ in $\Z$ when $n = p_1^{e_1} \cdots p_k^{e_k}$. By the Lasker–Noether theorem, every ideal of a Noetherian ring has one, and its associated primes are unique. Geometrically, it splits an algebraic set into irreducible components, with extra embedded components that record finer structure.