Multiplicative Derivations on Rank-s Matrices for Relatively Small s

Abstract

In this paper, we describe multiplicative derivations on the set of all rank-[Formula: see text] matrices of [Formula: see text] over a field [Formula: see text] with a relatively small integer [Formula: see text]. Concretely, for fixed integers [Formula: see text], [Formula: see text] satisfying [Formula: see text] and [Formula: see text], we prove that if a map [Formula: see text] satisfies [Formula: see text] for any two rank-[Formula: see text]matrices [Formula: see text], [Formula: see text], then there exists a derivation [Formula: see text] of [Formula: see text] such that [Formula: see text] for each rank-[Formula: see text] matrix [Formula: see text] with [Formula: see text]. As an application, we prove that a multiplicative derivation on a special subset of [Formula: see text] must be a derivation.