2024/12/07 by Lee, Tsiu-Kwen, Lin, Jheng-Huei
#16K40 (Secondary) #16N60 (Primary) 16W10 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2412.05588
Let R be a prime ring with center Z(R) and with involution *. Given an additive subgroup A of R, let T(A):=\x+x^*| x∈ A\ and K0(A):=\x-x^*| x∈ A\. Let L be a non-abelian Lie ideal of R. It is proved that if d is a nonzero derivation of R satisfying d(T(L))=0 (resp. d(K0(L))=0), then T(R)2⊆ Z(R) (resp. K0(R)2⊆ Z(R)). These results are applied to the study of d(T(M))=0 and d(K0(M))=0 for noncentral *-subrings M of a division ring R such that M is invariant under all inner automorphisms of R, and for noncentral additive subgroups M of a prime ring R containing a nontrivial idempotent such that M is invariant under all special inner automorphisms of R. The obtained theorems also generalize some recent results on simple artinian rings with involution due to M. Chacron.