---- Assume $k=\overline{k}$ is an [[algebraically closed]] [[field]]. > [!definition] Definition. ([[coordinate ring]]) > If $X$ is an [[algebraic set|(affine) algebraic set]], and $I(X)$ is the [[ideal]] of all [[polynomial 4|polynomials]] that vanish on $X$, then the [[quotient ring]] $k[X]:=k[T_{1},\dots,T_{n}] / I(X)$ is called the **coordinate ring** of $X$. > $R$ is regarded as the [[ring]] of polynomial functions on $X$. It is a [[subalgebra generated by a subset|finitely generated]] [[nilpotent element of a ring|reduced]] $k$-[[algebra]].[^1] Elements of $k[X]$ form the **[[ring]] of regular functions on $X$**; global sections of the structure [[sheaf]] of $X$. > [[Hilbert's geometry-algebra correspondence|Note that]] if $X$ is an [[affine variety]], [[Hilbert's geometry-algebra correspondence|then]] $I(X)$ is [[prime ideal|prime]], so the [[coordinate ring]] is an [[integral domain]] in such a case. If $X=*$ is a point, then $I(X)$ is [[maximal ideal|maximal]], so the [[coordinate ring]] is a [[field]]. ^definition ---- #### [^1]: F.g. because quotient of a [[polynomial 4|polynomial ring]]. Reduced because $I(X)$ is [[radical of an ideal|radical]] ([[strong Nullstellensatz]]), and an [[ideal]] $I$ in a [[ring]] $R$ is [[radical of an ideal|radical]] iff the [[quotient ring]] $R / I$ is [[nilpotent element of a ring|reduced]]. 'identify two polynomials iff their difference vanishes on $X$,... i.e., if they are the same *on $X$*' ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```