A coordinate ring is a mathematical structure used in algebraic geometry to study geometric objects, such as varieties. It is formed from the set of polynomial functions defined on an algebraic variety, where the ring operations are addition and multiplication of these polynomials. The coordinate ring provides a way to translate geometric problems into algebraic ones. In a more formal sense, if X is an algebraic variety, its coordinate ring, denoted as k[X], consists of all polynomial functions on X with coefficients in a field k. This ring captures the essential properties of the variety and allows mathematicians to analyze its structure and relationships through algebraic methods.