Let G be an algebraic group acting on an affine variety V . Hilbert’s 14th problem asked whether in this situation the ring of invariants must be finitely generated. Nagata showed that this is not always the case. Thus the ring of invariants may not be isomorphic to the ring of functions on an affine variety. Nevertheless it is necessarily isomorphic to the ring of functions of an quasi-affine variety. More precisely we show: Let k be a field and R an integrally closed k-algebra.
Then the following properties are equivalent: