Module Amenability and Weak Module Amenability for second Dual of Banach algebras

In this paper we define the weak module amenability of a Banach algebra $\mathcal{A}$ which is a Banach module over another Banach algebra $\mathfrak{A}$ with compatible actions, and show that under some mild conditions weak module amenability of$\mathcal{A}^{∗∗}$ implies weak module amenability of $\mathcal{A}.$ Also among other results, we investigate the relation between module Arens regularity of a Banach algebra and module amenability of its second dual. As a consequence we prove that $\ell^1(S)$ is always weakly module amenable (as an $\ell^1(E)$-module), where $S$ is an inverse semigroup with an upward directed set of idempotents $E$.