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# involutory ring

# General Definition of a Ring with Involution

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

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

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

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.

Remark. Note that the traditional definition of an involution on a vector space is different from the one given here. Clearly, $*$ is bijective, so that it is an anti-automorphism. If $*$ is the identity on $R$, then $R$ is commutative.

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. $aa^{*}$ 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. 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.

## Mathematics Subject Classification

16W10*no label found*

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