A group acting on a ring by automorphisms and its invariant subring

Let $G$ be a group (Group and abelian group) and $C$ a commutative ring. An action of $G$ on $C$ by ring automorphisms is a left action $G\times C\to C$ (Left group actions, transitive actions, and faithful actions), written $(g,c)\mapsto g\cdot c$, such that for every $g\in G$ the map $c\mapsto g\cdot c$ is a ring homomorphism (Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$). Each such map is then automatically bijective, with inverse the map given by $g^{-1}$, since $g^{-1}\cdot (g\cdot c)=(g^{-1}g)\cdot c=c$ and likewise in the other order; so each $g$ acts as a ring automorphism, and in particular $g\cdot 1_C=1_C$ and $g\cdot 0_C=0_C$.

The invariant subring is

$C^G:=\{c\in C:g\cdot c=c\text{ for every }g\in G\}\mathrm{.}$

It is a subring of $C$ (Subring: a subset containing $1_R$ and closed under addition, additive inverses and multiplication). It contains $1_C$, because every $g$ acts as a unital ring homomorphism; and for $c,c^{\mathrm{\prime }}\in C^G$ and $g\in G$ one has $g\cdot (c+c^{\mathrm{\prime }})=g\cdot c+g\cdot c^{\mathrm{\prime }}=c+c^{\mathrm{\prime }}$, $g\cdot (-c)=-(g\cdot c)=-c$ and $g\cdot (c{c}^{\mathrm{\prime }})=(g\cdot c)(g\cdot c^{\mathrm{\prime }})=c{c}^{\mathrm{\prime }}$, so $C^G$ satisfies (T1) to (T4).

Actions by algebra automorphisms. When $C$ is a commutative $A$-algebra with structure map ${\eta }_C\text{\hspace{0.17em}⁣}:A\to C$ (Algebras over a commutative ring, central structure maps, and algebra homomorphisms), an action by $A$-algebra automorphisms is one in which every $g$ fixes ${\eta }_C(A)$ pointwise: $g\cdot {\eta }_C(a)={\eta }_C(a)$ for all $g\in G$ and $a\in A$. Then ${\eta }_C(A)\subseteq C^G$, so $C^G$ is an $A$-subalgebra of $C$ and $A\subseteq C^G\subseteq C$ whenever $A$ is a subring of $C$.

The trivial group. If $G=\{e\}$ then $e\cdot c=c$ for every $c$, so $C^G=C$.

• This agrees with the notation already in use for symmetric polynomials, and does not compete with it. Symmetric polynomials as the invariants of variable permutations lets $\sigma \in {\mathrm{Sym}}_n$ act on $R[x_1,\dots ,x_n]$ by $\sigma \cdot f(x_1,\dots ,x_n)=f(x_{\sigma (1)},\dots ,x_{\sigma (n)})$ and writes $R[x_1,\dots ,x_n{]}^{{\mathrm{Sym}}_n}$ for the fixed subset. That is exactly $C^G$ for $C=R[x_1,\dots ,x_n]$ and $G={\mathrm{Sym}}_n$: the same set, under the same notation, and the definition here is the general form of it.

• Only invariance is asked for, not any finiteness. $G$ may be infinite, and $C^G$ may then be small; the finiteness of $G$ is a hypothesis of the results about $C^G$, not part of this definition.