$I=(x-y^2z,x+y^2z)$ and $I=(x^2y,y^3z,xyz)$ generate $K[x,y,z]$. If not, find two that do. ($K$ is a field and $I$ is an ideal).
Now, my question might really be trivial but here it goes: I know that if they would generate it this would imply that the elements of K[x,y,z] could be written as linear combinations of the elements of the Ideals, but how can I know this? I mean K[x,y,z] has infinitely many elements how do I check or understand that an Ideal does indeed generate them all. And the same problem goes when the ideal is not given and I have to find it. How can I be sure that it is indeed the correct Ideal.
I am a total beginner in Algebra and I have always problems with this sort of questions regarding generating ideals. People seem to see it right away and I don't know if there is any way which I am not aware of.
Thanks very much for any sort of help.