I am self studying commutative algebra, and I have a question about an exercise I am working on. Let $I$ be generated by the minors of the matrix:
$$\begin{bmatrix} x & y & z\\ y & z & w\end{bmatrix}$$
That is, $I = \langle xz - y^2, xw-yz, yw - z^2\rangle$. One part of the exercise is to show there is an $k[x,w]$-module monomorphism from $R = k[x,y,z,w]/I \to k[s,t]$. However, since $R$ has a $k[x,w]$-basis $\{1,y,z\}$, it is a rank 3 free module and there can be no such monomorphism.
I am inclined to believe I am wrong, since the text of the exercise suggests that the rank 3 free module injects into a rank 1 free module, but I do not know where. Is my basis incorrect, or am I misunderstanding the rank of a module?