I think the proof of K[x] being a PID could apply here, although you can only get one direction of containment, which is what you want
Let be a (algebraically closed) field.
Exists a characterization for not trivial ideals that satisfy the following property:
If an ideal satisfy (*), then is it contained in a principal ideal?
The problem arises from an excercise of algebraic geometry that I tried to generalize.
I didn't mean the exact same proof. The proof that K[x] is a PID works because you can do an induction on the degree of the polynomial. In the multivariable case you can set up a lexicographic order on the by ordering the vector . This is Grobner basis theory, btw.