TITLE: Involutions on commutative rings
QUESTION [8 upvotes]: 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?

REPLY [3 votes]: 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.