For a locally compact group G let $L^1(G, ω)$ be a Beurling algebra. We characterize injectivity property of $L^1(G, ω), M(G, ω)$ and $L^1(G, ω)$ as a Banach $L^1(G, ω)$-Modules. This characterization is employed to find a necessary and sufficient condition for amenability of G. In the special case where ${ω_n}^∞_n=1$ is a sequence of weight functions on G we prove the same result for Fr´echet algebras $A(ω)$ and $B(ω)$.
