involutory ring

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

∈

R

:

1.

(

a

+

b

)

*

=

a

*

+

b

*

,

2.

(

a

⁢

b

)

*

=

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

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

ϕ

:

R

→

S

is a

ring homomorphism

which respects involutions. More precisely,

ϕ

⁢

(

a

*

R

)

=

ϕ

⁢

(

a

)

*

S

,

for any

⁢

a

∈

R

.

By abuse of notation, if we use

*

to denote both

*

R

and

*

S

, then we see that any *-homomorphism

ϕ

commutes with

*

:

ϕ

*

=

*

ϕ

.

Special Elements

An

element

a

∈

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

∈

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

∈

R

, and we write

b

=

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

∈

R

. A

trace element

b

is an element expressible as

a

+

a

*

for some

a

∈

R

, and written

b

=

tr

⁡

(

a

)

.

Let

S

be a subset of

R

, write

S

*

:=

{

a

*

∣

a

∈

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

e

⁢

R

=

f

⁢

R

(

principal ideals

generated by

e

and

f

are equal), then

e

=

f

. For if

e

⁢

a

=

f

⁢

f

=

f

for some

a

∈

R

, then

f

=

e

⁢

a

=

e

⁢

e

⁢

a

=

e

⁢

f

. Similarly,

e

=

f

⁢

e

. Therefore,

e

=

e

*

=

(

f

⁢

e

)

*

=

e

*

⁢

f

*

=

e

⁢

f

=

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

Generated on Fri Feb 9 18:44:38 2018 by

LaTeXML