What does it mean for a ring to have an involution? Are there any examples?

Question

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?

Answer 1

An involution is a map which is it's own inverse. For example consider the complex number with the complex conjugation as involution: It's easy to check that complex conjugation satisfies the properties of involution.

Answer 2

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.

Answer 3

An easy example is $(\mathbb C, *)$ where $*$ is the conjugate operator.