On maps preserving products of matrices

Let D be a division ring with characteristic different from 2, and let R = M-n(D). The first goal of this paper is to describe an additive map f : R -> R satisfying the identity f(x)f(y) = m for every x, y is an element of R such that xy = k, where m, k is an element of R are fixed invertible elements. Additionally, let M = M-n(C), the set of all n x n matrices with complex entries. We will describe a bijective linear map g : M -> M satisfying g(X) circle g(Y) = M whenever X circle Y = K for every X, Y is an element of M, where M, K is an element of M are fixed, and circle denotes the Jordan product. (C) 2018 Elsevier Inc. All rights reserved.