TITLE: What does it mean for a ring to have an involution? Are there any examples?
QUESTION [5 upvotes]: A ring $R$ is said to be a ring with an involution if there exists a mapping 
$*\colon R  \to  R$ such that for every $a, b \in R$: 

$a^{**} = a$,
$(a + b)^* = b^* +  a^*$,
$(ab)^* = b^*a^*$.

Can anyone please explain this definition with an example?

REPLY [10 votes]: An involution is a so-called anti-homomorphism (that's what that last condition expresses) that, if applied twice, gives the identity map. Note that your second condition is the same as $(a+b)^* = a^*+b^*$ – addition is commutative!
As a typical example, let $n$ be a natural number, let $A$ be any ring, and consider the ring of $n$-by-$n$ matrices over $A$. The transpose is an involution on that ring. Another classical example: on the ring of Hamilton quaternions, the map $a+bi+cj+dk \mapsto a-bi-cj-dk$ is an involution.
If the ring is commutative, then an involution is just a ring automorphism (an invertible homomorphism from the ring to itself) of order 2.