A note on automorphisms of prime rings

We prove algebraic generalizations of some results of M. Awami and A. B. Thaheem concerning the equation α + α ^-1 = β + β^-1 where α,β are automorphisms of C^∗-algebras. We show that if R is a prime ring of characteristic not equal to 2 and α,β are automorphisms of R satisfying α + β^-1α^-1β = β + β^-1 , then α = β or α = α^-1. We also prove a result on the commutativity of α and β.