Abstract: Let A be an algebra with unit 1. A map f:A→A is a weakly additive map if for every x, y∈A there exist t_x,y, s_x,y∈F such that f(x + y)=t_x,yf(x) + s_x,yf(y). We prove that under some conditions, if f is a commuting map, then there exists λ₀(x)∈A and a map λ₁ from A into Z(A) such that f(x)=λ₀(x)x + λ₁(x) for all x∈A. As an application, a class of commuting weakly additive maps on M_n(F) are characterized.

Key words: algebras, commuting maps, weakly additive maps