I'm doing an exercise with $$A = \mathbb{R}[x, y, z] / \langle x^2 + y^2 + z^2 - 1 \rangle \; .$$
We have $$N = \{ (xf, yf, zf) \}, M = \{ (f, g, h) \mid xf + yg + zh = 0 \}$$ as submodules of $A^3$. First I was asked to prove $N \cong A$ and $M \oplus N \cong A^3$. Both follow by considering the "dot product" by (x, y ,z) map, that is $(f, g, h) \to xf + yg +zh$, whose kernel is precisely $M$ and whose restriction to $N$ yields the first isomorphism. We thus construct an exact sequence with $M, A^3, A$ and this map, and since $A$ is free over itself the sequence splits, hence the second isomorphism (since $A \cong N$). I'm them asked to prove $M$ is not free. This is where I'm stuck. I don't know if I'm missing something obvious but I have no clue how to do this. Maybe any two elements are linearly dependent? Or is there some other useful property of free modules that I'm missing? Any help is appreciated.