Additive operators preserving idempotent matrices over fields and applications Original Research Article

Suppose F is a field of characteristic not 2. Let Mn(F) be the algebra of all n × n matrices over F, and let image2, image3, image−1, and image# be the semigroups of all additive operator on Mn(F) that preserve idempotence, preserve tripotence, preserve inverses of matrices, and preserve group inverses of matrices, respectively. The main result in this paper is that the semigroup image2 is generated by transposition, similarity, the operators X → Xτ for fixed arbitrary injective endomorphisms τ on F, and the operators X → σ(tr X) for fixed arbitrary additive maps σ from F to Mn(F) with σ(1) = O. As applications, we determine the structures of image3, image−1, and image# when the characteristic of F is also not 3.