Principal ideal of $\mathbb{C}[Z,\bar{Z}]$

Question

Let $I$ be an ideal of $\mathbb{C}[Z,\bar{Z}]$.

How to prove that $I$ is principal in $\mathbb{C}[Z,\bar{Z}]$ ?

It exists some simple criterion to say that an ideal will be principal or not?

Answer 1

Consider an ideal $I=(f_1,\dotsc, f_s)$ in a noetherian factorial ring and let $f$ be the greatest common divisor of the $f_i$.

Then $I$ is principal if and only if $f \in (f_1, \dotsc, f_s)$ holds.

In practice, working in the polynomial ring, figuring out whether this holds, you need Groebner bases.

Thus, in practice, you can also directly use the following equivalence:

$I$ is principal if and only if its reduced Groebner basis consists of one element.