Fixed-point subring

In algebra, the fixed-point subring formula_1 of an automorphism "f" of a ring "R" is the subring of the fixed points of "f":
More generally, if "G" is a group acting on "R", then the subring of "R":
is called the fixed subring or, more traditionally, the ring of invariants. In Galois theory, when "R" is a field and "G" is a group of field automorphisms, the fixed ring is a subfield called the fixed field of the automorphism group; see Fundamental theorem of Galois theory.
Along with a module of covariants, the ring of invariants is a central object of study in invariant theory. Geometrically, the rings of invariants are the coordinate rings of (affine or projective) GIT quotients and they play fundamental roles in the constructions in geometric invariant theory.
Example: Let formula_4 be a polynomial ring in "n" variables. The symmetric group "S""n" acts on "R" by permuting the variables. Then the ring of invariants "R""G" is the ring of symmetric polynomials. If a reductive algebraic group "G" acts on "R", then the fundamental theorem of invariant theory describes the generators of "R""G".
Hilbert's fourteenth problem asks whether the ring of invariants is finitely generated or not (the answer is affirmative if "G" is a reductive algebraic group by Nagata's theorem.) The finite generation is easily seen for a finite group "G" acting on a finitely generated algebra "R": since "R" is integral over "R""G", the Artin–Tate lemma implies "R""G" is a finitely generated algebra. The answer is negative for some unipotent groups.
Let "G" be a finite group. Let "S" be the symmetric algebra of a finite-dimensional "G"-module. Then "G" is a reflection group if and only if formula_5 is a free module (of finite rank) over "S""G" (Chevalley's theorem).
In differential geometry, if "G" is a Lie group and formula_6 its Lie algebra, then each principal "G"-bundle on a manifold "M" determines a graded algebra homomorphism (called the Chern–Weil homomorphism)
where formula_8 is the ring of polynomial functions on formula_9 and "G" acts on formula_8 by adjoint representation.