Multiplicative Isomorphisms of Triangular Algerbas

Let T be a triangular algebra and R′ be an arbitrary ring.A multiplicative isomorphism φ∶T→R′ is a bijective map and satisfys that φ(ab)=φ(a)φ(b) for a,b∈T.It is proved that every multiplicative isomorphism from T onto R′ is additive under a mild condition on T using the method of matrix.Moreover,we give a necessary and sufficient condition of multiplicative isomorphism.