Free module or finitely generated?

Question

Could someone help me about the module $K[x,y]/(x^2y^2)$, $K$ is a field. I must to prove if it is a free $K[x]$-module or finitely generated $K[x]$-module or both.

I try find a set of generators but it seems difficult..thanks a lot..

Answer 1

The $K$-algebra $K[x,y]/(x^2 y^2)$ is not a finitely generated $K[x]$-module. Indeed, the powers $\{1,y,y^2,\ldots\}$ cannot be in the $K[x]$-span of any finite set. Also, $K[x,y]/(x^2 y^2)$ is not a free $K[x]$-module, because it has torsion elements. For example, the element $y^2\in K[x,y]/(x^2 y^2)$ satisfies $x^2\cdot y^2=0$, but $y^2\ne 0$ in $K[x,y]/(x^2 y^2)$ and $x^2\ne 0$ in $K[x]$.