Conversely, starting from some closed (affine) variety formula_29, the closure of "V" in formula_1 is the projective variety called the of "V". If formula_31 defines "V", then the defining ideal of this closure is the homogeneous ideal of formula_32 generated by  for all "f" in "I". For example, if "V" is an affine curve given by, say, formula_34 in the affine plane, then its projective completion in the projective plane is given by formula_35  For various applications, it is necessary to consider more general algebro-geometric objects than projective varieties, namely projective schemes. The first step towards projective schemes is to endow projective space with a scheme structure, in a way refining the above description of projective space as an algebraic variety, i.e., formula_36 is a scheme which it is a union of ("n" + 1) copies of the affine "n"-space "kn". More generally, projective space over a ring "A" is the union of the affine schemes  in such a way the variables match up as expected. The set of closed points of formula_38, for algebraically closed fields "k", is then the projective space formula_36 in the usual sense. An equivalent but streamlined construction is given by the Proj construction, which is an analog of the spectrum of a ring, denoted "Spec", which defines an affine scheme. For example, if "A" is a ring, then  If "R" is a quotient of formula_41 by a homogeneous ideal "I", then the canonical surjection induces the closed immersion  Compared to projective varieties, the condition that the ideal "I" be a prime ideal was dropped. This leads to a much more flexible notion: on the one hand the topological space formula_43 may have multiple irreducible components. Moreover, there may be nilpotent functions on "X". Closed subschemes of formula_38 correspond bijectively to the homogeneous ideals "I" of formula_41 that are saturated; i.e., formula_46 This fact may be considered as a refined version of projective Nullstellensatz. We can give a coordinate-free analog of the above. Namely, given a finite-dimensional vector space "V" over "k", we let  where formula_48 is the symmetric algebra of formula_49.