Let $T{\geq}S{\geq}R$ be commutative rings. I'm trying to prove that if $T$ is integral over $S$ and $S$ is integral over $R$ then $T$ is integral over $R$.
Let $t$ be in $T$ so there exist $s_{0},...,s_{n-1}$ in $S$ with $t^n+s_{n-1}t^{n-1}+...+s_{0}=0$. Then $T$ is integral over $C=R[s_{n-1},..,s_{0}]$. Then my book says that $C[t]$ is finitely generated as a $C$ module. Can anyone explain/prove this? Thanks
One uses the following result: let $A$ be a subring of $B$ with $t\in B$. Then the following statements are equivalent: (1) The element $t$ is integral over $A$, (2) The ring A[t] is a finitely-generated $A$-module.
This is a non-trivial result. It is usually proved in books on commutative algebra at the beginning of the section "Integral ring extensions".