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