Ring of invariants

Summary

The ring of invariants of a group acting on a commutative ring is the subring of elements fixed by every element of the group. For a group acting linearly on a vector space, it is the ring of polynomial functions that do not change under the group. It is the central object of invariant theory. Whether it is finitely generated is the subject of Hilbert's fourteenth problem. Its ring-theoretic properties are the subject of results such as the Hochster–Roberts theorem. Geometrically, it is the coordinate ring of the quotient of the space by the group.