Fixed-point subring

In algebra, the fixed-point subring $R^{f}$ of an automorphism f of a ring R is the subring of R:

$R^{f}=\{r\in R\mid f(r)=r\}.$

Slightly more generally, if G is a subgroup of the automorphism group $\operatorname {Aut} (R)$ of R, then $R^{G}$, the intersection of $R^{g},\,g\in G$ is a subring called the subring fixed by G or, more commonly, the ring of invariants. A basic example appears in Galois theory; see Fundamental theorem of Galois theory.

Hilbert's fourteenth problem asks whether the ring of invariants is finitely generated (the answer is affirmative if G is a reductive algebraic group.)