Module contractibility for semigroup algebras

In this paper, we find the relationships between module contractibility of a Banach algebra and its ideals. We also prove that module contractibility of a Banach algebra is equivalent to module contractibility of its module unitization. Finally, we show that when a maximal group homomorphic image of an inverse semigroup S with the set of idempotents E is finite, the module projective tensor product `1 (S)⊗b` 1(E)` 1 (S) is ` 1 (E)-module contractible.