For my thesis, I'm defining affine $k$-algebras to be reduced, finitely generated $k$-algebras--each of which turns out to be isomorphic to the quotient of a polynomial ring by a radical ideal.
I'm looking for a similar characterisation of "projective $k$-algebras" (i.e. those which occur as coordinate rings of projective varieties). A first attempt was to use reduced, finitely generated, graded $k$-algebras. However, after some searching it seems that for example $k[X^2, X^3]$ is not isomorphic to the quotient of a graded polynomial ring by a homogeneous radical ideal, unless we change the grading: see here and here.
Is there an elegant way of tweaking this definition to enforce the correct behaviour?