Let $(R,\mathfrak m, k)$ be a Noetherian, complete, local ring. Let $E$ be an injective hull of $k$. difine for every $R$-module $M$ $$\mu_M:M\to Hom_R(Hom_R(M,E),E)$$ is natural $R$ homomorphis, such that $\mu_M(x)(f)=f(x)$ , $x\in M$ and $f\in Hom(M,E)$. suppose $M$ is an $R$-module and has noetherian submodule $N$ such that $\frac{M}{N}$ is an artinian $R$-module.I want to show that $\mu_M$ is isomorphism. I known that $\mu_M$ is always one one but I dont know how to show that is it onto? maybe the commutative diagram is helpful. $$\begin{array} A0 & {\longrightarrow} & N \stackrel{f}{\longrightarrow} & M \stackrel{g}{\longrightarrow} & \frac{M}{N} & {\longrightarrow} & 0 \\ & & \downarrow{h} & \downarrow{h'}& \downarrow{h''}\\ 0 & {\longrightarrow} & N^{**} \stackrel{f^{**}}{\longrightarrow} & M^{**} \stackrel{g^{**}}{\longrightarrow} & \frac{M}{N}^{**} & {\longrightarrow} & 0 \end{array}\\ $$ $N^{**}=Hom_R(Hom_R(N,E),E)$
Any hint or solution would be helpful.
Thank You