Does a ring of coordinates separate points?

Question

Let $G$ be a linear algebraic group over $\mathbb C$. Is it true that $\mathbb C [G]$ separates points?

Since it is almost like a polynomial ring and it is a coordinates ring this should be easy because no 2 points should have the same coordinates, but I am just wondering.

Thanks!

Answer 1

This has nothing to do with $G$ being a group, for any affine algebraic variety $X$ over $\Bbb C$, the coordinate ring $\Bbb C[X]$ separates points. This is simply because you can embed $X\subseteq\Bbb A^n$ in some affine space, and for any two points $p,q\in X$ you will find a polynomial $f\in\Bbb C[x_1,\ldots,x_n]$ with $f(p)=0$ and $f(q)\ne0$. Then, $f|_X\in\Bbb C[X]$ will have the same property.