2025/08/03 by Lee, Tsiu-Kwen
#16N60 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2508.01544
A prime ring R with extended centroid C is said to be exceptional if both \rm char R=2 and dimCRC=4. Herstein characterized additive subgroups A of a nonexceptional simple ring R satisfying [A, [R, R]]⊆ A. In 1972 Lanski and Montgomery extended Herstein's theorem to nonexceptional prime rings. In the paper we first extend Herstein's theorem to arbitrary simple rings. For the prime case, let R be an exceptional prime ring with center Z(R). It is proved that if A is a noncentral additive subgroup of R satisfying [A, L]⊆ A for some nonabelian Lie ideal L of R, then βZ(R)⊆ A for some nonzero β∈ Z(R), and either AC=Ca+C for some a∈ A∖ Z(R) with a2∈ Z(R) or [RC, RC]⊆ AC. Secondly, we study certain generalized linear identities satisfied by Lie ideals and then completely characterize derivations δ, d of R satisfying δd(L)⊆ Z(R) for L a Lie ideal of R.