2×2 矩阵代数保持幂等的映射(英文)

Maps on 2 × 2 matrix algebras preserving idempotence

Let M2 be the algebra of all 2 × 2 matrices over a ?eld F of characteristic 2 and |F| > 2. Let P2 be the set of all idempotent matrices in M2. It is shown that if φ : M2 → M2 is an injective map such that A ? λB ∈ P2 implies φ(A) ? λφ(B) ∈ P2 for any A,B ∈ M2 and λ ∈F, then either φ is of the form φ(A) = TAT? for any A ∈ M2, or φ is of 1 the form φ(A) = TAtT? for any A ∈ M2, where T ∈ M2 is a nonsingular matrix.

Let M2 be the algebra of all 2 × 2 matrices over a ?eld F of characteristic 2 and |F| > 2. Let P2 be the set of all idempotent matrices in M2. It is shown that if φ : M2 → M2 is an injective map such that A ? λB ∈ P2 implies φ(A) ? λφ(B) ∈ P2 for any A,B ∈ M2 and λ ∈F, then either φ is of the form φ(A) = TAT? for any A ∈ M2, or φ is of 1 the form φ(A) = TAtT? for any A ∈ M2, where T ∈ M2 is a nonsingular matrix. 1