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

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 (http://planetmath.org/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 $𝒃$, any linear transformation over $T$ (to itself) can be represented by a square matrix $M$ over $k$ via $𝒃$. The map taking $M$ to its transpose $M^T$ is an involution. If $k$ is $ℂ$, 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 $\varphi :R\to S$ is a ring homomorphism which respects involutions. More precisely,

$$\varphi (a^{{*}_R})=\varphi {(a)}^{{*}_S},\text{ for any }a\in R.$$

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

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. $a{a}^{*}$ 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 $a{a}^{*}$ for some $a\in R$, and we write $b=\mathrm{n}(a)$. If $a{a}^{*}=a^{*}a$, 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 $b=\mathrm{tr}(a)$.

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.

Title	involutory ring
Canonical name	InvolutoryRing
Date of creation	2013-03-22 15:41:01
Last modified on	2013-03-22 15:41:01
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	32
Author	CWoo (3771)
Entry type	Definition
Classification	msc 16W10
Synonym	ring admitting an involution
Synonym	involutary ring
Synonym	involutive ring
Synonym	ring with involution
Synonym	Hermitian element
Synonym	symmetric element
Synonym	self-adjoint
Synonym	adjoint
Synonym	projection
Synonym	involutive ring
Related topic	HollowMatrixRings
Defines	involution
Defines	adjoint element
Defines	self-adjoint element
Defines	projection element
Defines	norm element
Defines	trace element
Defines	skew symmetric element
Defines	*-homomorphism
Defines	normal element
Defines	unitary element