Le théorème de Hua pour les algèbres artiniennes simples

Abstract

Let D be a division ring and M n ( D) be the ring of the n× n matrices with entries in D. Consider a surjective mapping σ : M n(D)→M n(D) satisfying σ( A+ B)= σ( A)+ σ( B) for all A,B∈M n(D), σ(1)=1 and for all invertible A in M n(D), σ(A) is invertible and σ( A −1)= σ( A) −1. If n=1 the well-known Hua's theorem states that σ is an automorphism or an anti-automorphism. We show that if D≠ F 2 (the field of two elements) then σ is an automorphism or an anti-automorphism for all n.