A subset of an algebraic variety which is itself a variety. Every variety is a subvariety of itself; other subvarieties are called
proper subvarieties.

defines the sphere of radius 1 centered at the origin. Its intersection with the -plane
is a circle given by the system of polynomial equations:

(2)

(3)

Hence the circle is itself an algebraic
variety, and a subvariety of the sphere, and of the plane as well.

Whenever some new independent equations are added to the equations defining a certain variety, the resulting variety will be smaller, since its points will be subject
to more conditions than before. In the language of ring
theory, this means that, while the sphere is the zero set of all polynomials
of the ideal of , every subvariety of it will be defined by a larger
ideal; this is the case for , which is the defining ideal
of the circle.