On additive transformations preserving a multiplicative matrix function

摘要

Let Mn(K) be the ring of all n×n matrices over a division ring K, and f be a multiplicative matrix function from Mn(K) to a multiplicative Abelian group with zero G∪{0} (f(AB)=f(A)f(B),∀A,B∈Mn(K)). We call an additive transformation ϕ on Mn(K) preserves a multiplicative matrix function f, if f(ϕ(A))=f(A),∀A∈Mn(K). In this paper, we characterize all additive surjective transformations on Mn(K) over any division ring K (chK≠2) that leave a non-trivial multiplicative matrix function invariant. Applications to several related preservers are considered.