Centralizing Maps in Prime Rings with Involution Original Research Article

Let R be a prime ring with involution, of characteristic ≠ 2, with center Z, skew elements K, and extended centroid C. THEOREM. Suppose [K, K] ≠ {0} and ƒ: K → Kis an additive map such that [ƒ(x), x] set membership, variant Z for all x set membership, variant K. Then, unless R is an order in a 16-dimensional central simple algebra, there exists λ set membership, variant C and an additive map μ: K → C such that ƒ(x) = λx + μ(x) for all x set membership, variant K.