Involutions on commutative rings I found that all the commutative rings with involution I know are the following:


*

*complex number with complex conjugation (plus similar constructions based on rationals and its extensions),

*any commutative ring with trivial involution,

*direct sum of two copies of a ring with involution * (not necessary nontrivial) with involution which sends pair $(a,b)$ to $(b^*,a^*)$,

*direct sum of involutionary rings with involution acting component-wise.


My question is: are there any other commutative rings with involution?
 A: Let $X$ be a set and let $\sigma:X\to X$ be an involution. If $k$ is a ring, then there is an induced involution $\sigma:R\to R$ in the ring $R=k[X]$ of all functions $X\to k$.
By restricting this general situation, you get new examples. For example, 


*

*if $X$ is a topological space, $\sigma:X\to X$ is continuous, $k=\mathbb C$ and $R=C(X)$ is the ring of all continuous real functions on $X$;

*if $X$ is a manifold, $\sigma:X\to X$ is differentiable, and $R=C^\infty(X)$ is the ring of all smooth real functions on $X$;

*&c.
In a sense, all examples are of this nature. Indeed, let $R$ be a comm. ring and let $\sigma:R\to R$ be an involution. If $X=\mathrm{Spec}\;R$ is the sprectrum of $R$, then there is an induced morphism $\sigma^*:X\to X$ and we recover the action of $\sigma$ on $R$ by looking at the action of $\sigma^*$ on the ring $\mathscr{O}_X$ of global sections of the structure sheaf on $X$.
A: The situation turns on whether $R$ contains nonzero solutions of $x+x=0$ (2-torsion) and whether $x+x=y$ can be solved for all $y$ (2-divisibility).
If $R$ admits division by $2$ then there is an additive decomposition $x =\frac{x + \sigma(x)}{2} + \frac{x - \sigma(x)}{2}$ as a sum of invariant and anti-invariant parts with respect to the involution.  $R$ is an extension of its invariant subring $R_0$ by a set of elements on which the involution acts as $-1$.
Otherwise, consider $\Bbb{Z}[X,Y]$ and $\Bbb{Z}/2[X,Y]$ each with the involution exchanging $X$ and $Y$.  In the first case there are subrings of symmetric and anti-symmetric functions but sums of those do not fill the whole ring. In the second case there is no canonical projection onto the invariant subring. 
A: For $X$ any topological space, the ring of continuous functions $X \to R$ where $R$ is any topological commutative ring with involution is itself a commutative ring with pointwise involution (consider in particular the case $R = \mathbb{C}$ with the usual topology). For $X$ compact Hausdorff we get important examples of $C^{\ast}$-algebras.
Note also that if $k$ is any field, any quadratic extension $L/k$ is a commutative ring with involution where the involution is the unique nontrivial automorphism of $L$ as a $k$-extension. 
Really there are many examples and many ways of constructing examples. 
