higher homomorphisms and ‎their approximations

In this paper, we introduce a class of higher homomorphisms on an algebra a and we characterize the structure of them as a linear combination of some sequences of homomorphisms. also, we prove that for any approximate higher ring homomorphism on a banach algebra a under some sequences of control funtions, there exists a unique higher ring homomorphism near it. using special sequences of control functions, we show that the approximate higher ring homomorphism is an exact higher ring homomorphism.