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# Article

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Keywords:
Banach modules; module derivation; module amenability; inverse semigroup
Summary:
Let $S$ be an inverse semigroup with the set of idempotents $E$ and $S/\approx$ be an appropriate group homomorphic image of $S$. In this paper we find a one-to-one correspondence between two cohomology groups of the group algebra $\ell ^1(S)$ and the semigroup algebra ${\ell ^{1}}(S/\approx )$ with coefficients in the same space. As a consequence, we prove that $S$ is amenable if and only if $S/\approx$ is amenable. This could be considered as the same result of Duncan and Namioka [5] with another method which asserts that the inverse semigroup $S$ is amenable if and only if the group homomorphic image $S/\sim$ is amenable, where $\sim$ is a congruence relation on $S$.
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