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Let $𝒜$ be a dual Banach algebra. We investigate the first weak${}^{*}$-continuous cohomology group of $𝒜$ with coefficients in $𝒜$. Hence, we obtain conditions on $𝒜$ for which $$H_{w^{*}}^1(𝒜,𝒜)=\left\{0\right\}.$$

Let $𝒜$ be a Banach algebra. $𝒜$ is called ideally amenable if for every closed ideal $I$ of $𝒜$, the first cohomology group of $𝒜$ with coefficients in $I^{*}$ is zero, i.e. $H^1(𝒜,I^{*})=\left\{0\right\}$. Some examples show that ideal amenability is different from weak amenability and amenability. Also for $n\in N$, $𝒜$ is called $n$-ideally amenable if for every closed ideal $I$ of $𝒜$, $H^1(𝒜,I^{\left(n\right)})=\left\{0\right\}$. In this paper we find the necessary and sufficient conditions for a module extension Banach algebra to be 2-ideally amenable.