First module cohomology group of induced semigroup algebras

‎Let $S$ be a discrete semigroup and $T$ be a left multiplier operator on $S$‎. ‎A new product on $S$ defined by $T$ creates a new induced semigroup $S _{T} $‎. ‎In this paper‎, ‎we show that if $T$ is bijective‎, ‎then the first module cohomology groups $ \HH_{\ell^1(E)}^{1}(\ell^1(S)‎, ‎\ell^{\infty}(S))$ and $ \HH_{\ell^1(E_{T})}^{1}(\ell^1({S_{T}})‎, ‎\ell^{\infty}(S_{T})) $ are equal‎, ‎where $E$ and $E_{T}$ are sets of idempotent elements in $S$ and $S _{T}$‎, ‎respectively‎. ‎Which in particular means that $\ell^1(S)$ is weak $\ell^1(E)$-module amenable if and only if $\ell^1(S_T)$ is weak $\ell^1(E_T)$-module amenable‎. ‎Finally‎, ‎by giving an example‎, ‎we show that the condition of bijectivity for $T$‎, ‎is necessary‎.