2024/10/04 by Manju, Praveen, Sharma, Rajendra Kumar
#FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2410.16279
In this article, we study twisted derivations of cyclic group rings. Let R be a commutative ring with unity, G be a finite cyclic group, and (σ, τ) be a pair of R-algebra endomorphisms of the group algebra RG, which are R-linear extensions of the group endomorphisms of G. In this article, we give two characterizations concerning (σ, τ)-derivations of the group ring RG. First, we develop a necessary and sufficient condition for a (σ, τ)-derivation of RG to be inner. Second, we provide a necessary and sufficient condition for an R-linear map D: RG → RG with D(1) = 0 to be a (σ, τ)-derivation. We also illustrate our theorems with the help of examples. As a consequence of these two characterizations, we answer the well-known twisted derivation problem for RG: Under what conditions are all (σ, τ)-derivations of RG inner? Or is the space of outer (σ, τ)-derivations trivial? More precisely, we give a sufficient condition under which all (σ, τ)-derivations of RG are inner and a sufficient condition under which RG has non-trivial outer (σ, τ)-derivations. Our result helps in generating several examples of non-trivial outer derivations.