On maps preserving products of matrices

Abstract

Let D be a division ring with characteristic different from 2, and let R=Mn(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∈R such that xy=k, where m,k∈R are fixed invertible elements. Additionally, let M=Mn(C), the set of all n×n matrices with complex entries. We will describe a bijective linear map g:M→M satisfying g(X)∘g(Y)=M whenever X∘Y=K for every X,Y∈M, where M,K∈M are fixed, and ∘ denotes the Jordan product.