Let R* be a ring with unit element I. For every A ∈ R*, let the mapping A → A′ be an involutory anti-automorphism, i.e. a one-to-one mapping of R* onto itself with the following properties:

Let C* denote the set of all symmetric elements of R*, i.e. C′ = C for all C ∈ C*.

An element A ∈ R* is said to be skew-symmetric if A′ + A = 0 (the zero element of the ring).