Lasker–Noether theorem

Remarks

For existence, call an ideal irreducible if it is not the intersection of two strictly larger ideals. By the Noetherian property, every proper ideal is a finite intersection of irreducible ideals, and in a Noetherian ring every irreducible ideal is primary. Removing redundant terms and joining terms with the same radical gives a minimal decomposition. For example, in $k[x, y]$ the ideal $(x^2, xy)$ has the minimal decompositions $(x) \cap (x^2, y)$ and $(x) \cap (x^2, xy, y^2)$. In both, the primes are $(x)$ and $(x, y)$, and the component $(x)$ at the minimal prime is the same, while the embedded component changes. Without the Noetherian hypothesis, existence can fail. In the product ring $R = \prod_{i \in \N} \F_2$, every prime ideal is maximal and there are infinitely many, such as the kernels of the projections. If $0 = Q_1 \cap \cdots \cap Q_n$ with $Q_i$ primary, then every prime contains some $\sqrt{Q_i}$, so $R$ would have at most $n$ prime ideals.

Going deeper

For a student

The primes in the theorem are the associated primes, and the minimal ones among them are the minimal primes over $I$. For an ideal of a polynomial ring, they describe the irreducible components of the zero set and the embedded points. The same theorem holds for submodules of finitely generated modules over a Noetherian ring. The uniqueness of the isolated components comes from localization: $Q_i$ is the preimage of $I R_P$ in $R$, for $P = \sqrt{Q_i}$ minimal.