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involutory ring (Definition)

General Definition of a Ring with Involution

Let $ R$ be a ring. An involution $ *$ 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 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 $ \boldsymbol{b}$, any linear transformation over $ T$ (to itself) can be represented by a square matrix $ M$ over $ k$ via $ \boldsymbol{b}$. The map taking $ M$ to its transpose $ M^T$ is an involution. If $ k$ is $ \mathbb{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 $ \phi:R\to S$ is a ring homomorphism which respects involutions. More precisely,

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

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

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. $ 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 $ a\in R$, and we write $ b=\operatorname{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 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=\operatorname{tr}(a)$.

Let $ S$ be a subset of $ R$, write $ S^*:=\lbrace a^*\mid a\in S\rbrace$. 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.



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See Also: hollow matrix rings

Other names:  ring admitting an involution, involutary ring, involutive ring, ring with involution, Hermitian element, symmetric element, self-adjoint, adjoint, projection, involutive ring
Also defines:  involution, adjoint element, self-adjoint element, projection element, norm element, trace element, skew symmetric element, *-homomorphism, normal element, unitary element
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Cross-references: linear algebra, skew symmetric, symmetry, characteristic, generated by, principal ideals, idempotent, subset, trace, addition, normal, unitary, multiplicative inverse, expressible, norm, ring homomorphism, composition, conjugate transpose, transpose, map, square matrix, linear transformation, basis, field, finite dimensional, module, rings of endomorphisms, commutative, identity, anti-automorphism, bijective, vector space, identity map, square, anti-endomorphism, ring
There are 69 references to this entry.

This is version 29 of involutory ring, born on 2006-02-16, modified 2007-06-22.
Object id is 7625, canonical name is InvolutaryRing.
Accessed 10435 times total.

Classification:
AMS MSC: 16W10 (Associative rings and algebras :: Rings and algebras with additional structure :: Rings with involution; Lie, Jordan and other nonassociative structures)
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