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单词 InvolutoryRing
释义

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,bR:

  1. 1.

    (a+b)*=a*+b*,

  2. 2.

    (ab)*=b*a*,

  3. 3.

    a**=a

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 (http://planetmath.org/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 , then the map taking M to its conjugate transposeMathworldPlanetmath M¯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.

*-Homomorphisms

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

ϕ(a*R)=ϕ(a)*S, for any aR.

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 aR 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 aR,

  1. 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 aR, and we write b=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 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 aR. A trace element b is an element expressible as a+a* for some aR, and written b=tr(a).

Let S be a subset of R, write S*:={a*aS}. 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 eR=fR (principal idealsMathworldPlanetmathPlanetmath generated by e and f are equal), then e=f. For if ea=ff=f for some aR, 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 symmetryPlanetmathPlanetmathPlanetmath. An element a in R is skew symmetric if a=-a*. Again, the name of this is borrowed from linear algebraMathworldPlanetmath.

Titleinvolutory ring
Canonical nameInvolutoryRing
Date of creation2013-03-22 15:41:01
Last modified on2013-03-22 15:41:01
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id32
AuthorCWoo (3771)
Entry typeDefinition
Classificationmsc 16W10
Synonymring admitting an involution
Synonyminvolutary ring
Synonyminvolutive ring
Synonymring with involution
SynonymHermitian element
Synonymsymmetric element
Synonymself-adjoint
Synonymadjoint
Synonymprojection
Synonyminvolutive ring
Related topicHollowMatrixRings
Definesinvolution
Definesadjoint element
Definesself-adjoint element
Definesprojection element
Definesnorm element
Definestrace element
Definesskew symmetric element
Defines*-homomorphism
Definesnormal element
Definesunitary element
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更新时间:2025/5/4 21:36:18