Originally Posted by

**robeuler** Let $\displaystyle R=C[x,y]$ the ring of polynomials in two variables with complex coefficients. Let $\displaystyle M=(x,y) $ be the ideal generated by $\displaystyle x$ and $\displaystyle y$. $\displaystyle M$ is an $\displaystyle R$-module.

Is $\displaystyle M$ a free module?

I am still in the habit of associating free with finitely generated, but I think the notion is more subtle than that. My instinct says yes: something along the lines of elements in $\displaystyle M$ looking like

$\displaystyle m=r_{1}x+r_{2}y$