In this paper, we introduce the new notion module Johnson amenabil-ity for a Banach algebra which is a Banach module over another Banach algebra with compatible actions. We study the relations between this new notion and other various notions of module amenability. We characterize the module Johnson amenability of ` 1 (S) as an ` 1 (E)-module, for an inverse semigroup S with subsemigroup E of idempotents. We investigate the module Johnson amenability of ` 1 (S), whenever S is a Brandt semigroup or bicyclic semigroup or N with maximum as its product. As application we show that for every non-empty set Λ, MΛ(C) as an A-module is module Johnson amenable if and only if Λ is finite, where A = [ai,j ] ∈ MΛ(C) | ∀i =6 j, ai,j = 0 .
