involutory ring
General Definition of a Ring with Involution
Let $R$ be a ring. An $*$ on $R$ is an antiendomorphism whose square is the identity map. In other words, for $a,b\beta \x88\x88R$:

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

2.
${(a\beta \x81\u2019b)}^{*}={b}^{*}\beta \x81\u2019{a}^{*}$,

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 antiautomorphism. 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 $\mathrm{\pi \x9d\x92\x83}$, any linear transformation over $T$ (to itself) can be represented by a square matrix^{} $M$ over $k$ via $\mathrm{\pi \x9d\x92\x83}$. The map taking $M$ to its transpose^{} ${M}^{T}$ is an involution. If $k$ is $\mathrm{\beta \x84\x82}$, then the map taking $M$ to its conjugate transpose^{} ${\stackrel{{\rm B}\u2015}{M}}^{T}$ 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^{} $\mathrm{{\rm O}\x95}:R\beta \x86\x92S$ is a ring homomorphism which respects involutions. More precisely,
$$\mathrm{{\rm O}\x95}\beta \x81\u2019({a}^{{*}_{R}})=\mathrm{{\rm O}\x95}\beta \x81\u2019{(a)}^{{*}_{S}},\text{\Beta for any\Beta}\beta \x81\u2019a\beta \x88\x88R.$$ 
By abuse of notation, if we use $*$ to denote both ${*}_{R}$ and ${*}_{S}$, then we see that any *homomorphism $\mathrm{{\rm O}\x95}$ commutes with $*$: $\mathrm{{\rm O}\x95}*=*\mathrm{{\rm O}\x95}$.
Special Elements
An element $a\beta \x88\x88R$ such that $a={a}^{*}$ is called a selfadjoint^{}. A ring with involution is usually associated with a ring of square matrices over a field, as such, a selfadjoint element is sometimes called a Hermitian element, or a symmetric element. For example, for any element $a\beta \x88\x88R$,

1.
$a\beta \x81\u2019{a}^{*}$ and ${a}^{*}\beta \x81\u2019a$ are both selfadjoint, the first of which is called the norm of $a$. A norm element $b$ is simply an element expressible in the form $a\beta \x81\u2019{a}^{*}$ for some $a\beta \x88\x88R$, and we write $b=\mathrm{n}\beta \x81\u2018(a)$. If $a\beta \x81\u2019{a}^{*}={a}^{*}\beta \x81\u2019a$, 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 selfadjoint elements $a+{a}^{*}={a}^{*}+a$, called the trace of $a$, for any $a\beta \x88\x88R$. A trace element $b$ is an element expressible as $a+{a}^{*}$ for some $a\beta \x88\x88R$, and written $b=\mathrm{tr}\beta \x81\u2018(a)$.
Let $S$ be a subset of $R$, write ${S}^{*}:=\{{a}^{*}\beta \x88\pounds a\beta \x88\x88S\}$. Then $S$ is said to be selfadjoint if $S={S}^{*}$.
A selfadjoint that is also an idempotent^{} in $R$ is called a projection. If $e$ and $f$ are two projections in $R$ such that $e\beta \x81\u2019R=f\beta \x81\u2019R$ (principal ideals^{} generated by $e$ and $f$ are equal), then $e=f$. For if $e\beta \x81\u2019a=f\beta \x81\u2019f=f$ for some $a\beta \x88\x88R$, then $f=e\beta \x81\u2019a=e\beta \x81\u2019e\beta \x81\u2019a=e\beta \x81\u2019f$. Similarly, $e=f\beta \x81\u2019e$. Therefore, $e={e}^{*}={(f\beta \x81\u2019e)}^{*}={e}^{*}\beta \x81\u2019{f}^{*}=e\beta \x81\u2019f=f$.
If the characteristic of $R$ is not 2, we also have a companion concept to selfadjointness, 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  20130322 15:41:01 
Last modified on  20130322 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  selfadjoint 
Synonym  adjoint 
Synonym  projection 
Synonym  involutive ring 
Related topic  HollowMatrixRings 
Defines  involution 
Defines  adjoint element 
Defines  selfadjoint element 
Defines  projection element 
Defines  norm element 
Defines  trace element 
Defines  skew symmetric element 
Defines  *homomorphism 
Defines  normal element 
Defines  unitary element 