The coordinate ring of an affine variety 
in the 
-dimensional affine
space 
,
where 
is an algebraically closed field,
is the quotient ring
![K[V]=K[x_1,...,x_n]/I(V),](/images/equations/CoordinateRing/NumberedEquation1.svg) |
where 
is the ideal of all polynomials
with coefficients in 
that vanish at every point of 
. If 
is the entire affine space 
, this ideal
is the zero ideal, so the coordinate ring is the polynomial ring ![K[x_1,...,x_n]](/images/equations/CoordinateRing/Inline10.svg)
. For a plane curve
defined by the Cartesian equation 
in the affine plane 
, with 
a nonzero polynomial, the coordinate
ring of its zero locus is ![K[x_1,x_2]/sqrt((f))](/images/equations/CoordinateRing/Inline14.svg)
, where 
is the ideal radical
of 
.
It equals ![K[x_1,x_2]/(f)](/images/equations/CoordinateRing/Inline17.svg)
when 
already equals its ideal radical (Milne 2024, pp. 47-48).
In general, the Krull dimension of ![K[V]](/images/equations/CoordinateRing/Inline19.svg)
equals the dimension of 
as a closed
set in the Zariski topology of 
. Two polynomials 
and 
define the same function on 
iff 
. Hence the elements
of ![K[V]](/images/equations/CoordinateRing/Inline26.svg)
are equivalence classes identified with the
polynomial functions from 
to 
.
More generally, the coordinate ring of an affine scheme 
is the ring
of global sections 
of its structure
sheaf. For 
,
it is naturally isomorphic to 
(Stacks Project, Tag 01HV). Unlike the coordinate ring of
an affine variety, it need not be a reduced
ring. For example, ![K[x]/(x^2)](/images/equations/CoordinateRing/Inline33.svg)
contains a nonzero nilpotent element, the class
of 
,
whereas the coordinate ring of its reduced zero locus is ![K[x]/(x)=K](/images/equations/CoordinateRing/Inline35.svg)
. A nilpotent element
maps to zero at every point valued in a field,
so point values alone do not recover the coordinate ring of a nonreduced affine
scheme.
