# 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?

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The ring of rational functions of one variable under $X\mapsto 1/X$? Or $R[X]$ under $X\mapsto -X$? –  Henning Makholm Oct 5 '11 at 21:08
@Henning's examples fit in Qiaochu's example of quadratic extensions very nicely. –  Mariano Suárez-Alvarez Oct 5 '11 at 21:13
@HenningMakholm. Oh, shame on me for not noticing such obvious examples. –  andrei Oct 6 '11 at 8:44

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 $X$.

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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.

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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.