Localization of Gorenstein ring

Question

Let $R$ be a Gorenstein local ring and $S=R \setminus Z(R)$. I want to prove $S^{-1}R =⊕_{ht\ p=0} R_p$ and $S^{-1}R$ is injective $R$-module.
I can see the above $p$'s are minimal, $id_{R_p} R_p=0$ and $S^{-1}R =⊕_{htp=0} E(R/p)$. And now?
thanks

Answer 1

Since $R$ is Gorenstein, $R$ is Cohen-Macaulay, so the minimal primes are exactly the associated primes. The localization maps $S^{-1}R \to R_p$ for $\text{ht}(p) = 0$ give a map $\displaystyle S^{-1}R \to \bigoplus_{\text{ht}(p)=0} R_p$, which is locally an isomorphism. Then note that $R_p$ is injective as an $R_p$-module, and thus (since injective hulls localize) also as an $R$-module, and direct sums of injectives over a Noetherian ring are again injective.