Primary decomposition

Remarks

Example: in $k[x, y]$ over a field $k$, $(x^2, xy) = (x) \cap (x^2, y)$, where $(x)$ is prime and $(x^2, y)$ is $(x, y)$-primary; the radicals differ and neither term contains the other, so the decomposition is minimal. Non-example: $(6) = (2) \cap (6)$ in $\Z$ is not a primary decomposition, because $(6)$ is not primary. The decomposition $(x^2, xy) = (x) \cap (x^2, xy, y^2)$ is another minimal one, so the primary components need not be unique. In the same way one defines a primary decomposition of a submodule $N$ of a module $M$, with primary submodules.