A note on maps preserving products of matrices

Abstract

Let $ D $ be a division ring such that either char$ (D)\neq 2, 3 $ or $ D $ is not a field and char$ (D)\neq 2 $. Let $ R = M_n(D) $ be the matrix ring over $ D $, where $ n > 1 $. Let $ m, k $ be fixed invertible elements in $ R $. The main purpose of the paper is to give a description of a bijective additive map $ f $: $ R\rightarrow R $, satisfying the identity $ f(x)f(y) = m $ for every $ x, y\in R $ with $ xy = k $, which gives a correct version of a result due to Catalano et al. in 2019.

Keywords

• maps preserving products
• jordan isomorphism
• division ring
• matrix ring