Maps Preserving Idempotence on Matrix Spaces

Abstract EN

Suppose F is an arbitrary field.

Let | F | be the number of the elements of F .

Let M n ( F ) be the space of all n × n matrices over F , let S n ( F ) be the subset of M n ( F ) consisting of all symmetric matrices, and let T n ( F ) be the subset of M n ( F ) consisting of all upper-triangular matrices.

Let V ∈ { S n ( F ) , M n ( F ) , T n ( F ) } ; a map Φ : V → V is said to preserve idempotence if A - λ B is idempotent if and only if Φ ( A ) - λ Φ ( B ) is idempotent for any A , B ∈ V and λ ∈ F .

In this paper, the maps preserving idempotence on S n ( F ) , M n ( F ) , and T n ( F ) were characterized in case | F | = 3 .