Let $n$ be a positive integer and let $k$ be an algebraically closed field. What is the coordinate ring of $GL(n,k)$ (the set of all $n \times n$ matrices with entries in $k$)? Here we identify this set as a subset of $k^{n^{2}}$.
Would it suffice to say that the coordinate ring is the localization of $k[x_{11},x_{12},..,x_{nn}]$ at the determinant function? Is there a way to "simplify" this?