Abstract :
Let R be a prime ring with the extended centroid C. An (anti)automorphism g of R is said to be Frobenius if, in the case of char(R) = ∞, = αg = α for all α set membership, variant C, or in the case of char(R) = p ≥ 0, αg = αpn for all α set membership, variant C, where n is a fixed integer, which may be zero, positive, or negative. (Anti)automorphisms g1, g2, … are said to be strongly independent if for any i ≠ j, gig−1j is not Frobenius. The following is proved: MAIN THEOREM. Let ψ(zijk) be a polynomial with Frobenius (anti)automorphisms. Assume that ψ(xΔjgki) = 0 holds on a nonzero two-sided ideal of R, where xi are distinct indeterminates, Δj are distinct regular derivation words (with respect to a linearly ordered basis of outer derivations), and gk are strongly independent (anti)automorphisms of R. Then ψ(zijk) = 0 holds on the two-sided Utumi quotient ring of R. If ψ(zijk) does not involve antiautomorphisms, then ψ(zijk) = 0 also holds on the left Utumi quotient ring of R.
