Module Amenability of Banach Algebras

Let $\mathfrak{A}$ and $\mathcal{A}$ be Banach algebras, and let $\mathcal{A}$ be a Banach $\mathfrak{A}$-bimodule. In this paper, at first we generalize some theorems from amenable Banach algebras into module amenable Banach algebras. We show that when $\mathcal{A}$ and $I$ are commutative Banach $\mathfrak{A}$-bimodules, and $\mathcal{A}$ is module amenable, where $I$ is two-sided closed ideal in $\mathcal{A}$, then $I$ is module amenable. By this, we show that if $I$ is a two sided ideal in an amenable inverse semigroup $S,$ then $I$ is amenable.