Let $R$ be a Noetherian ring, $W$ a multiplicative subset of $R$, and $M$ an $R$-module. Show that $E_{W^{-1}R}(W^{-1}M) \cong W^{-1}E_R(M)$ as $W^{-1}R$-modules.
I want to use the following facts:
Let $z$ an element of $W^{-1}M$. Show that there exists an element $x$ of $M$ so that $W^{-1}R\dfrac{x}{1}=W^{-1}Rz$ and $ \operatorname{ann}_R x= \operatorname{ann}_{W^{-1}R} z\cap R$ (i.e. $Rx\hookrightarrow W^{-1}R\dfrac{x}{1}=W^{-1}Rz$)
If $M\subset E$ is an essential extension of $R$-modules then $W^{-1}M\subset W^{-1}E$ is an essential extension of $W^{-1}R$-modules.
How do I do this?