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Let R be a commutative ring, G a group and RG its group ring. Let φ: RG → RG denote the R-linear extension of an involution φ defined on G. An element x in RG is said to be antisymmetric if φ(x) = -x. A characterization is given of when the antisymmetric elements [Formula: see text] of RG commute except when Char(R) = 3.

Let R be a commutative ring with unity and let G be a group. The group ring RG has a natural involution that maps each element g ∈ G to its inverse. We denote by RG − the set of skew symmetric elements under this involution. We study necessary and sufficient conditions for RG − to be commutative.

Let R be a commutative ring, G a group and RG its group ring. Let ϕ : RG → RG denote the R-linear extension of an involution ϕ defined on G. An element x in RG is said to be ϕ-antisymmetric if ϕ(x) = −x. A characterization is given of when the ϕ-antisymmetric elements of RG commute. This is a completion of earlier work.

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