We introduce two notions of amenability for a Banach algebra A. Let n ∈ N and I be a closed two-sided ideal in A. A is n − I−weakly amenable if the first cohomology group of A with coefficients in the n-th dual space I (n) is zero; i.e., H 1 (A, I (n)) = {0}. Further, A is n-ideally amenable if A is n−I−weakly amenable for every closed two-sided ideal I in A. We find some relationships of n − I− weak and m − J− weak amenabilities for some different m and n or for different closed ideals I and J of A.