I'm trying to learn about Linear Algebra in Rings. It's a new topic for me, and I'm coming across some definitions I don't understand and would appreciate the help.
So, I understand that a module is free if it has a basis. For example, $R^n$ is always free because we can find a basis to generate the vector space.
However, the following module $M$ isn't: $R = C[x,y]$ and $M$ be the ideal generated by $x$ and $y$.
Elements in the ideal can be expressed as a linear combination of the elements that generate it, correct?
So aren't these elements acting like a basis, or what am I missing?
They are not linearly independent over $\mathbb{C}[x,y]$. Take, for example, the linear combination $(-y)x + (x) y=0$. In this example, the elements generate $M$, but they are not a basis.