Commuting maps on rank-1 matrices over noncommutative division rings

ABSTRACT

Let n≥3 be a natural number. Let M_n(𝔻) be the ring of all n×n matrices over a noncommutative division ring 𝔻. In the present paper, we will find the description of all additive mappings $G:M_n\left(𝔻\right)\to M_n\left(𝔻\right)$ such that [G(y),y] = G(y)y−yG(y) = 0 for all rank-1 matrix y. Precisely, we will prove that G(x) = λx+μ(x) for all x∈M_n(𝔻), where λ lies in the center of 𝔻 and μ is a central map.

KEYWORDS:

• Commuting maps
• division rings
• functional identities
• rank-1 matrices

AMS SUBJECT CLASSIFICATION:

• 16R60