Reynolds operator on a polynomial invariant ring ($\mathcal{R}(f)={\int }_{K}k\cdot fdk$)

In characteristic zero, averaging over a compact group, or over a Zariski-dense compact form of a complex reductive group, gives a degree-preserving projection onto the polynomial invariant ring. Its module property is $\mathcal{R}(af)=a\mathcal{R}(f)$ for invariant $a$. This property turns finite ideal generation in the ambient polynomial ring into finite algebra generation of the invariant subring.