Weighted Jordan homomorphisms.

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 c T (x 2) = T (x) 2 for all x ∈ A. We provide assumptions, which are in particular fulfilled when A = B = M n (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) ≠ 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 (x 2) = T (x) 2 for all x ∈ A. [ABSTRACT FROM AUTHOR]