# involutory ring

## General Definition of a Ring with Involution

Let $R$ be a ring. An $*$ on $R$ is an anti-endomorphism whose square is the identity map. In other words, for $a,b\in R$:

1. 1.

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

2. 2.

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

3. 3.

$a^{**}=a$

A ring admitting an involution   is called an involutory ring. $a^{*}$ is called the adjoint  of $a$. By (3), $a$ is the adjoint of $a^{*}$, so that every element of $R$ is an adjoint.

Examples. Involutory rings occur most often in rings of endomorphisms over a module. When $V$ is a finite dimensional vector space over a field $k$ with a given basis $\boldsymbol{b}$, any linear transformation over $T$ (to itself) can be represented by a square matrix  $M$ over $k$ via $\boldsymbol{b}$. The map taking $M$ to its transpose  $M^{T}$ is an involution. If $k$ is $\mathbb{C}$, then the map taking $M$ to its conjugate transpose  $\overline{M}^{T}$ is also an involution. In general, the composition of an isomorphism      and an involution is an involution, and the composition of two involutions is an isomorphism.

## *-Homomorphisms

Let $R$ and $S$ be involutory rings with involutions $*_{R}$ and $*_{S}$. A *-homomorphism   $\phi:R\to S$ is a ring homomorphism which respects involutions. More precisely,

 $\phi(a^{*_{R}})=\phi(a)^{*_{S}},\quad\mbox{ for any }a\in R.$

By abuse of notation, if we use $*$ to denote both $*_{R}$ and $*_{S}$, then we see that any *-homomorphism $\phi$ commutes with $*$: $\phi*=*\phi$.

## Special Elements

An element $a\in R$ such that $a=a^{*}$ is called a self-adjoint  . A ring with involution is usually associated with a ring of square matrices over a field, as such, a self-adjoint element is sometimes called a Hermitian element, or a symmetric element. For example, for any element $a\in R$,

1. 1.

$aa^{*}$ and $a^{*}a$ are both self-adjoint, the first of which is called the norm of $a$. A norm element $b$ is simply an element expressible in the form $aa^{*}$ for some $a\in R$, and we write $b=\operatorname{n}(a)$. If $aa^{*}=a^{*}a$, then $a$ is called a normal element. If $a^{*}$ is the multiplicative inverse of $a$, then $a$ is a unitary element. If $a$ is unitary  , then it is normal.

2. 2.

With respect to addition, we can also form self-adjoint elements $a+a^{*}=a^{*}+a$, called the trace of $a$, for any $a\in R$. A trace element $b$ is an element expressible as $a+a^{*}$ for some $a\in R$, and written $b=\operatorname{tr}(a)$.

Let $S$ be a subset of $R$, write $S^{*}:=\{a^{*}\mid a\in S\}$. Then $S$ is said to be self-adjoint if $S=S^{*}$.

A self-adjoint that is also an idempotent  in $R$ is called a projection. If $e$ and $f$ are two projections in $R$ such that $eR=fR$ (principal ideals   generated by $e$ and $f$ are equal), then $e=f$. For if $ea=ff=f$ for some $a\in R$, then $f=ea=eea=ef$. Similarly, $e=fe$. Therefore, $e=e^{*}=(fe)^{*}=e^{*}f^{*}=ef=f$.

If the characteristic of $R$ is not 2, we also have a companion concept to self-adjointness, that of skew symmetry   . An element $a$ in $R$ is skew symmetric if $a=-a^{*}$. Again, the name of this is borrowed from linear algebra  .