Let $R\subset T$ be two commutative rings, and $T$ is integral over $R$. Let $\mathfrak m\in \operatorname{Max} R,\mathfrak n\in\operatorname{Max}T$ such that $\mathfrak m=\mathfrak n\cap R$. Show that $R_\mathfrak m\subset T_\mathfrak n$, i.e. the canonical map $\{\frac ru\mid r\in R,u\in R\setminus\mathfrak m\}\to\{\frac tv\mid t\in T,v\in T\setminus\mathfrak n\}$ must be an injection.
I feel confused because the exercise 4 from Chapter 5 in the book of Atiyah and Macdonald just assume that the localization is a ring extension. And I don't know if the condition "integral over" is necessary.
Best Answer
Let $R=\mathbb Z_4$ be the ring of residue classes modulo $4$, and $T=\mathbb Z_4[X]/(X^2+X,2X)$. Denote by $x$ the residue class of $X$ modulo the ideal $(X^2+X,2X)$. Then $T=R[x]$.
since $x$ is integral over $R$.
The ideal $\mathfrak n=(2,x+1)$ is maximal in $T$ since $T/\mathfrak n\simeq\mathbb Z/2\mathbb Z$. Moreover, $\mathfrak n\cap R=\mathfrak m$, where $\mathfrak m=2\mathbb Z_4$.
In order to see this note that $\frac 21\ne\frac 01$ in $R_{\mathfrak m}$, but $\frac 21=\frac 01$ in $T_{\mathfrak n}$ since $2x=0$, and $x\in T\setminus\mathfrak n$.