# Thread: Ideals in polynomials ring

1. ## Ideals in polynomials ring

Let $K$ be a (algebraically closed) field.
Exists a characterization for not trivial ideals $I \subset K[x_1,...,x_n]$ that satisfy the following property:
(*) $\forall f,g \in I, \ gcd(f,g) \ne 1$.

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.

2. 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

3. Of course, if $n=1$ the claim is obviously true since $K[x]$ is a PID.
But I asked in the case $n > 1$, and in this case $K[x_1,...,x_n]$ is not a PID.

4. 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 $\prod_{i=1}^n x_i^{c_i}$ by ordering the vector $(c_1,...,c_n)$. This is Grobner basis theory, btw.