Coincidence of the coordinate ring and the set of regular functions If $X$ is an affine variety, and if we denote by k[X] its coordinate ring and by $O(X)$ the set of regular functions on $X$, then it is well-known that $k[X]\cong O(X)$.
My question is : Do we have the same for (affine) algebraic groups ? How do we prove that?
 A: Well, affine algebraic groups are in particular affine varieties, so of course the same result holds for them.
For non-affine algebraic groups, the ring of regular functions can be very restrictive, and the coordinate ring of a non-affine variety is not really well defined.  For example, a connected abelian variety $G$ is complete (that is, for any variety $Y$, the projection map $G \times Y \rightarrow Y$ is closed), and therefore has trivial global section.
Definitions: if $X$ is a variety, $\mathcal O(X)$ denotes the ring of regular functions on $X$, i.e. morphisms of varieties from $X$ to the variety $k = \mathbb A^1$.
If $X$ is an affine variety, and $X$ is a Zariski closed subset of some $k^n$, then the coordinate ring $k[X]$ is the ring of polynomial functions from $k^n$ to $k$, restricted to $X$.  Equivalently, $X$ is can be identified with the space of maximal ideals (or prime ideals, depending on the formulation) of a finitely generated $k$-algebra $A$, and the coordinate ring $k[X]$ is just $A$.
