A homogeneous ideal in a graded ring is an ideal generated by a set of homogeneous elements, i.e., each one is contained in only one of the . For example, the polynomial ring is a graded ring, where . The ideal , i.e., all polynomials with no constant or linear terms, is a homogeneous ideal in . Another homogeneous ideal is in .
Given any finite set of polynomials in variables, the process of homogenization converts them to homogeneous polynomials in variables. If is a polynomial of degree then
is the homogenization of . Similarly, if is an ideal in , then is its homogenization and is a homogeneous ideal. For example, if then . Note that in general, if then may have more elements than . However, if , ..., form a Gröbner basis using a graded monomial order, then . A polynomial is easily dehomogenized by setting the extra variable .
The affine variety corresponding to a homogeneous ideal has the property that iff for all complex . Therefore, a homogeneous ideal defines an algebraic variety in complex projective space.