The twisted derivation problem for group rings

We study $$(\sigma ,\tau )$$ -derivations of a group ring RG where G is with center having finite index in and R semiprime 1 such that either has no torsion elements or if p-torsion elements, then p does not divide the order let $$\sigma $$ be R-linear endomorphisms fixing pointwise. generalize Main Theorem 1.1 Chaudhuri (Comm Algebra 47(9): 3800–3807, 2019) prove there $$T\supset R$$ $$\mathcal {Z}(T)\supset \mathcal {Z}(R)$$ for natural extensions , \tau to TG, we get $$H^1(TG,{}_\sigma TG_\tau )=0$$ $${}_\sigma twisted $$TG-TG$$ -bimodule. provide applications above result (2019) integral rings groups connect derivations other important problems field as isomorphism problem Zassenhaus conjectures. also give an example which both locally nilpotent every F, exists F-linear -derivation FG -inner.