Check whether submodule is free

Question

Let $R= \mathbb{C}[X,Y]$ and $M=\langle X,Y\rangle$ (ideal generated by $X$ and $Y$). Is $M$ a free $R$-module?

Answer 1

A free $R$-submodule of $R$ has at most rank 1. Let $M \subseteq R$ be a free $R$-submodule of $R$. Assume there exists a basis containing $f,g\in M$, then we have

$$ (-g)\cdot f + g\cdot f = 0.$$

Thus, $f$ and $g$ are $R$-linearly dependent, which contradicts the assumption that $f,g$ are part of a basis (Note that this argument works for any ring, as we didn't use any properties of $R$).

Hence, as $\langle X, Y \rangle$ is not generated by a single element, you can conclude that it is not a free $R$-submodule.

Answer 2

Its just $Y\cdot X + (-X)\cdot Y = 0$ which needs to be considered.