TITLE: Primary decomposition of a monomial ideal
QUESTION [5 upvotes]: Can anyone give me an idea about the primary decomposition of the ideal $I=(x^3y,xy^4)$ of the ring $R=k[x,y]$? 

I am trying to connect the primary decomposition with the set Ass(R/I) which i have find before..

REPLY [9 votes]: 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}