On additive transformations preserving a multiplicative matrix function (original) (raw)

Let M n ðKÞ be the ring of all n  n matrices over a division ring K, and f be a multiplicative matrix function from M n ðKÞ to a multiplicative Abelian group with zero G [ f0g ðf ðABÞ ¼ f ðAÞf ðBÞ; 8A; B 2 M n ðKÞÞ. We call an additive transformation / on M n ðKÞ preserves a multiplicative matrix function f, if f ð/ðAÞÞ ¼ f ðAÞ; 8A 2 M n ðKÞ. In this paper, we characterize all additive surjective transformations on M n ðKÞ over any division ring K ðchK 6 ¼ 2Þ that leave a non-trivial multiplicative matrix function invariant. Applications to several related preservers are considered.