Suppose $K$ is a field and consider the ideal $(x^2, xy, xz, yz)$ of $K[x, y, z]$. Find a primary decomposition of $(x^2, xy, xz, yz)$.
I have read about a strategy of finding primary decomposition using $\gcd$ of generators as follow:
Using $\gcd(y, z)=1$, we can get $$(x^2, xy, xz, yz)=(x^2, xy,xz, y) \cap (x^2,xy,xz, z)=(x^2, xz, y)\cap(x^2,xy,z)$$ Since $\gcd(x, z)=1$, $$(x^2,xz,y)=(x^2, x, y)\cap (x^2,y,z)=(x,y) \cap (x^2, y, z)$$ and similarly $$(x^2,xy,z)=(x,z)\cap(x^2,y,z)$$ Now put everything together and get $$(x^2,xy,xz, yz)=(x,y)\cap (x,z)\cap(x^2,y,z)$$ It seems all ideals are now primary. Am I applying the strategy correctly?