σ-Approximately module amenable Banach algebras

In this paper, we define the notion of sigma-approximate module amenability of Banach algebras and give some properties about this notion. Also for Banach (mathfrak {A})-bimodule (mathcal {A}), and (mathcal {J}), the closed ideal of (mathcal {A}) generated by elements of form ((alpha cdot a) b-a(bcdot alpha )), ((a,bin mathcal {A}text {, }quad alpha in mathfrak {A}) ), we considered some corollaries about (widehat{sigma })-approximate amenability of (frac{mathcal {A}}{mathcal {J}}) as a Banach (mathcal {A})-bimodule, where (widehat{sigma }:frac{mathcal {A}}{mathcal {J}}rightarrow frac{mathcal {A}}{mathcal {J}}) by (widehat{sigma }(a+mathcal {J}) =sigma (a) +mathcal {J}) has a dense range.