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?

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 $X$.