Primary decomposition of a monomial ideal
If $I$ is a monomial ideal with a generator $ab$ (where $a$ and $b$ are coprime), say $I = (ab) + J$, then $I = ((a) + J) \cap ((b) + J)$. Applying this recursively gives a primary decomposition for any monomial ideal:
\begin{align} I &= (x^3y, xy^4) = (x^3, xy^4) \cap (y, xy^4) \\ &= (x^3, x) \cap (x^3, y^4) \cap (y, x) \cap (y, y^4) \\ &= (x) \cap (x^3, y^4) \cap (x, y) \cap (y) \\ &= (x) \cap (x^3, y^4) \cap (y) \end{align}