General Definition of a Ring with Involution
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 , then is commutative.
Examples. Involutory rings occur most often in rings of endomorphisms over a module. When is a finite dimensional vector space over a field with a given basis , any linear transformation over (to itself) can be represented by a square matrix over via . The map taking to its transpose is an involution. If is , then the map taking to its conjugate transpose 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.
By abuse of notation, if we use to denote both and , then we see that any *-homomorphism commutes with : .
An element such that 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 ,
With respect to addition, we can also form self-adjoint elements , called the trace of , for any . A trace element is an element expressible as for some , and written .
Let be a subset of , write . Then is said to be self-adjoint if .
A self-adjoint that is also an idempotent in is called a projection. If and are two projections in such that (principal ideals generated by and are equal), then . For if for some , then . Similarly, . Therefore, .
If the characteristic of is not 2, we also have a companion concept to self-adjointness, that of skew symmetry. An element in is skew symmetric if . Again, the name of this is borrowed from linear algebra.
|Date of creation||2013-03-22 15:41:01|
|Last modified on||2013-03-22 15:41:01|
|Last modified by||CWoo (3771)|
|Synonym||ring admitting an involution|
|Synonym||ring with involution|
|Defines||skew symmetric element|