I'm looking for help understanding the coordinate ring. The definition I have roughly says There is a surjection PI:k[t_1 , ... , t_n] ---> k[X] given by restricting a polynomial to X, where X is an algebraic set. Then k[X] is called the coordinate ring. As i understand it all this is saying is take any polynomial in k[t_1 , ... , t_n] can now only be evaluated with elements in X. So its like changing the domain of the polynomials, (this change is just a shrinking of the bigger n-dimensional space hence the word restriction) If my understanding is right, then why is it called the coordinate ring? does this name have a special meaning/derivation?