Does the module $M=\langle x^2+y^2 \rangle$ have a basis over $K[x,y]$

Question

How would I show that the module $M=\langle x^2+y^2 \rangle$ does or doesn't have a basis over $K[x,y]$, where $K$ is a field. Would the set $\{x^2+y^2\}$ be considered a basis for $M$.

Answer 1

A principal ideal is a free module if and only if the generator is not a zero-divisor.

On the other hand for any ideal to be free, it is necessary that it is principal. So principal ideals generated by non-zero divisors (regular elements) are precisely the ideals, which admit a basis as a module.