  We extend the concept of amenability of a Banach algebra $A$ to the case that
there is an extra $\mathfrak A$-module structure on $A$, and show that when $S$
is an inverse semigroup with subsemigroup $E$ of idempotents, then
$A=\ell^1(S)$ as a Banach module over $\mathfrak A=\ell^1(E)$ is module
amenable iff $S$ is amenable. When $S$ is a discrete group, $\ell^1(E)=\mathbb
C$ and this is just the celebrated Johnson's theorem.
