Weighted Jordan homomorphisms

Abstract

Let A and B be unital rings. An additive map T : A → B is called a weighted Jordan homomorphism if c = T (1) is an invertible central element and cT (x2) = T (x)2 for all x ∈ A. We provide assumptions, which are in particular fulfilled when A = B = Mn(R) with n ≥ 2 and R any unital ring with 1 2 , under which every surjective additive map T : A → B with the property that T (x)T (y) + T (y)T (x) = 0 whenever xy = yx = 0 is a weighted Jordan homomorphism. Further, we show that if A is a prime ring with char(A) 6= 2, 3, 5, then a bijective additive map T : A → A is a weighted Jordan homomorphism provided that there exists an additive map S : A → A such that S(x2) = T (x)2 for all x ∈ A.