Let $R$ be an Artinian ring, and let $\mathfrak p$ and $\mathfrak q$ be distinct prime ideals of $R$. I have to prove that $(R_{\mathfrak p})_{\mathfrak q}=0$.
What I have done is the following: Since $R$ is Artinian we have $\dim R=0$, so every prime ideal of $R$ is maximal, so $\mathfrak p\not\subset\mathfrak q$ and vice versa. If we localize at $\mathfrak p$ we invert all elements outside of $\mathfrak p$, so we invert at least one element of $\mathfrak q$. Then $\mathfrak qR_{\mathfrak p}$ contains a unit and thus equals $R_{\mathfrak p}$, so if we localize at $\mathfrak qR_{\mathfrak p}$ we invert everything outside $\mathfrak qR_{\mathfrak p}=R_{\mathfrak p}$, so we invert seemingly invert..nothing. This doesn't make sense.
Can somebody offer some help? It is much appreciated.
In order to prove this we want to show that $\frac01\in S$, that is, there is $a\in R\setminus\mathfrak q$ and $b\in R\setminus\mathfrak p$ such that $ab=0$.
Denote by $\mathfrak m_1,\dots,\mathfrak m_t$ the maximal ideals of $R$ others that $\mathfrak p$ and $\mathfrak q$. Now we can pick $x\in\mathfrak p\setminus\mathfrak q$ and $y\in(\mathfrak q\cap\mathfrak m_1\cap\cdots\cap\mathfrak m_t)\setminus\mathfrak p$. Then $xy\in\mathfrak p\cap\mathfrak q\cap\mathfrak m_1\cap\cdots\cap\mathfrak m_t$, so it is nilpotent, that is, there is $k\ge1$ such that $x^ky^k=0$. Now set $a=x^k$ and $b=y^k$.