Characterisation of "projective $k$-algebras" 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?
 A: In other words, you want to know for which graded $k$-algebras $S$ the associated projective scheme $\operatorname{Proj} S$ is in fact a variety.
The usual scheme-theoretic definition of a (possibly reducible) variety over $k$ is a reduced, separated scheme of finite type over $k$.  Since projective morphisms are automatically of finite type and separated (at least for noetherian schemes) [1], we just need $\operatorname{Proj S}$ to be reduced and projective over $k$.
To ensure that $S$ is reduced and projective over $k$, it's enough [2] that $S$ be reduced, $S_0 = k$, and $S$ finitely generated by $S_1$.
This condition isn't quite necessary for $\operatorname{Proj} S$ to be a projective scheme, I think -- for instance, I believe you could just take a graded ring and double all the gradings without affecting the associated scheme -- but I believe (someone should probably check this) that every projective variety arises from such a ring.
[1] Hartshorne, Theorem II.4.9
[2] Hartshorne, Example II.4.8.1
