Anticommutativity of Skew-symmetric Elements under Generalized Oriented Involutions

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Let R be a ring with char(R)≠2 whose unit group are denoted by U(R), G a group, and RG its group ring. Let * be an involution in G, σ:G(R) be a nontrivial group homomorphism, with kerσ=N, satisfying xx*∈ N for all x∈ G, and define the generalized oriented involution σ* in RG by (∑ₓ∈ Gαₓx)σ*=∑ₓ∈ Gσ(x)αₓx*. An element α∈ RG is called skew-symmetric if ασ*=-α, and the set of all skew-symmetric elements are denoted by (RG)-. In this paper, we will classify the group rings RG such that (RG)- is anticommutative, generalizing, and obtaining as consequence, the main result of GP13a.