I'm doing this exercise (from the book of Bosch):
Let $R$ be an Artinian ring and let $\mathfrak{p}_1, \ldots \mathfrak{p}_n $ be its (pairwise different) prime ideals. Show that:
a) The canonical homomorphism $R \to \prod_{i = 1}^{r} R/\mathfrak{p}_i^n $ is an isomorphism if $n$ is large enough.
b) The canonical homomorphisms $R_{\mathfrak{p}_i} \to R/\mathfrak{p}_i^n$, $\ i = 1, \ldots, r $ are isomorphisms if $n$ is large enough.
I've done point a) but I'm stuck on point b). Any hint?
Let $\mathfrak m$ be a maximal ideal of a commutative ring $R$. Define $f:R_{m}\to R/m^k$ by $f(a/s)=\hat a\hat s^{-1}$. (Note that $s\in R-m$, so $\hat s$ is invertible in $R/m^k$ which is a local ring with maximal ideal $m/m^k$.) Obviously $f$ is surjective. Moreover, $$a/s\in\ker f\iff\hat a\hat s^{-1}=\hat 0\iff\hat a=\hat 0\iff a\in m^k,$$ and therefore $\ker f=m^kR_m$. If $m^kR_m=(0)$, $f$ is injective, that is, an isomorphism.