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. 1.


  2. 2.


  3. 3.


A ring admitting an involutionPlanetmathPlanetmathPlanetmath is called an involutory ring. a* is called the adjointPlanetmathPlanetmath 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 spaceMathworldPlanetmath is different from the one given here. Clearly, * is bijectiveMathworldPlanetmathPlanetmath, so that it is an anti-automorphism. If * is the identityPlanetmathPlanetmath on R, then R is commutativePlanetmathPlanetmathPlanetmath.

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 matrixMathworldPlanetmath M over k via 𝒃. The map taking M to its transposeMathworldPlanetmath MT is an involution. If k is β„‚, then the map taking M to its conjugate transposeMathworldPlanetmath MΒ―T is also an involution. In general, the composition of an isomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath and an involution is an involution, and the composition of two involutions is an isomorphism.


Let R and S be involutory rings with involutions *R and *S. A *-homomorphismPlanetmathPlanetmathPlanetmath ϕ:R→S is a ring homomorphism which respects involutions. More precisely,

ϕ⁒(a*R)=ϕ⁒(a)*S,Β for any ⁒a∈R.

By abuse of notation, if we use * to denote both *R and *S, then we see that any *-homomorphism Ο• commutes with *: Ο•*=*Ο•.

Special Elements

An element a∈R such that a=a* is called a self-adjointPlanetmathPlanetmath. 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. 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 unitaryPlanetmathPlanetmath, 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∈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 idempotentPlanetmathPlanetmath in R is called a projection. If e and f are two projections in R such that e⁒R=f⁒R (principal idealsMathworldPlanetmathPlanetmath 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 symmetryPlanetmathPlanetmathPlanetmath. An element a in R is skew symmetric if a=-a*. Again, the name of this is borrowed from linear algebraMathworldPlanetmath.

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