TITLE: What is a primary decomposition of the ideal $I = \langle xy, x - yz \rangle$?
QUESTION [20 upvotes]: Given the ring $k[x,y,z]$, where $k$ is a field, and an ideal $I=(xy,x-yz)$, find a primary decomposition of $I$. 

I tried to draw the graph of the variety of $I$ and get a decomposition of $(x,y)\cap(x,z)$ (the two prime ideals corresponds to the irreducible components of the variety), but apparently these two ideals are not the primary decomposition of $I$, and I don't know how to fix this.
I would really appreciate it if someone can help me. thx

REPLY [5 votes]: If you are planning to work more on ring theory, I recommend installing Macaulay2 on your computer. It is a computer algebra software for computing with rings and ideals and modules.
For example, to compute the primary decomposition of your ideal, one would type the following commands:
R = QQ[x,y,z]
I = ideal (x*y, x-y*z)
primaryDecomposition I

{ideal (z, x), ideal (y*z - x, y^2 , x*y)}

This actually disagrees with Jim Belk's answer above.