On (？？; ¯)-skew-commuting and (？？; ¯)-skew-centralizing maps in rings with left identity

Abstract

Let R be a ring with left identity. Let G : R£R ! R be a symmetric biadditive mapping and g the trace of G. Let ?? : R ! R be an endomorphism and ¯ : R ! R an epimorphism. In this paper we show the following: (i) Let R be 2- torsion-free. If g is (??; ¯)-skew-commuting on R, then we have G = 0. (ii) If g is (¯; ¯)-skew -centralizing on R, then g is (¯; ¯)-commuting on R. (iii) Let n ¸ 2. Let R be (n+1)!-torsion- free. If g is n-(??; ¯)-skew-commuting on R, then we have G = 0. (iv) Let R be 6-torsion-free. If g is 2-(??; ¯)-commuting on R, then g is (??; ¯)-commuting on R.