Let $R = k[u, v, x, y]$ and $I = (ux, vy, uy + vx)$. What will be the primary decomposition of $I$?
My try:
The ideal $(u,v)$ and $(x,y)$ is the minimal prime ideal of $I$. And so the radical $\sqrt{I}=(u,v)\cap (x,y)$. So certainly they are minimal associated primes of $I$.
Now, what will be the other embedded primes? Also, what will be primary decomposition of $I$? Thanks in advance.