  In the present paper, the concepts of module (uniform) approximate
amenability and contractibility of Banach algebras that are modules over
another Banach algebra, are introduced. The general theory is developed and
some hereditary properties are given. In analogy with the Banach algebraic
approximate amenability, it is shown that module approximate amenability and
contractibility are the same properties. It is also shown that module uniform
approximate (contractibility) amenability and module (contractibility,
respectively) amenability for commutative Banach modules are equivalent.
Applying these results to $\ell^1(S)$ as an $\ell^1(E)$-module, for an inverse
semigroup $S$ with the set of idempotents $E$, it is shown that $\ell^1(S)$ is
module approximately amenable (contractible) if and only if it is module
uniformly approximately amenable if and only if $S$ is amenable. Moreover,
$\ell^1(S)^{**}$ is module (uniformly) approximately amenable if and only if a
maximal group homomorphic image of $S$ is finite.
