Module-Amenability on Module Extension Banach Algebras

Let A be a Banach algebra and E be a Banach A-bimodule then S = A  E, the l1-direct sum of A and E becomes a module extension Banach algebra when equipped with the algebras product (a; x):(a?; x?) = (aa?; a:x? + x:a?). In this paper, we investigate ?-amenability for these Banach algebras and we show that for discrete inverse semigroup S with the set of idempotents ES, the module extension Banach algebra S = l1(ES)  l1(S) is ?-amenable as a l1(ES)-module if and only if l1(ES) is amenable as Banach algebra.