16 - Skew Fields with Involution

Summary

Let A be a ring (with 1 ≠ 0), then we call as usual a Z-module automorphism θ of A a ring antiautomorphism of A if θ(x)θ(y) = = θ(yx) for all x, y ∈ A. In this context the following is obvious:

1. (1) If θ is a ring antiautomorphism of A, then θ defines an isomorphism A ≃ A^op.

2. (2) If θ is a ring antiautomorphism of A, then the same is true for θ⁻¹.

3. (3) If θ, Ω are ring antiautomorphisms of A, then θΩ is a ring automorphism of A.

4. (4) If θ is a ring antiautomorphism of A, then, for any a ∈ A^* θ_a: A → A, x ↦ aθ(x)a⁻¹ is again a ring antiautomorphism of A.

5. (5) Any ring antiautomorphism θ of A can be extended to a ring antiautomorphism of M_n(A) via θ(x_ij) ≔ (θ(x_ji)).

Definition 1.Let A be a ring (with 1 ≠ 0) and I a ring antiautomorphism of A such that I² = id_A, then I is called an “involution of A” and we write^Ix in place of I(x) (x ∈ A). Moreover, we denote by S_I(A) ≔ {x ∈ A|^Ix = x} the Z-module of “I-symmetric elements” in A.