Theorem: if $R,S$ are integral domains, $R\subset S$, and where $s_{1},...,s_{n}$ in $R_{S}$. Then there is a $m\in \mathbb{N}$ and $t_{1},..., t_{m}$ in $S$ (not all 0) so that $s_{i}N\subset N $ ($i=1,...,n)$ where $N=t_{1}R+...t_{m}R$
I was ill yesterday and could not attend the lecture (where it was shown). Does anybody know where I can find a proof of this theorem on the internet or a book? Or does anybody know how to show this and is willing to write it here? Thanks.
I will post now the proof that Milne gives for another Theorem but I fail to use for the theorem of my lecturer:
Proposition 5.1. Let A be a subring of a ring B. An element $\alpha$ of B is integral over A if and only if there exists a faithful (i.e. if $a M = 0 $ then this implies $a=0$) $A[\alpha]$ submodule M of B that is finitely generated as an A-module.
Proof $\Rightarrow$ : Suppose $\alpha ^{n} + a_{1}\alpha^{n-1}+\cdots+a_{n}=0 $
Then the A-submodule M of B generated by $1,\alpha,\ldots,\alpha^{n-1}$ has the property that $\alpha M \subset M$, and it is faithful because it contains 1 .
$\Leftarrow : $ Let M be an A-module in B with a finite set $\{e_{1},\ldots,e_{n}\}$ of generators such that $\alpha M \subset M $ is faithful as an $A[\alpha]$ module. Then, for each i, $\alpha e_{i} = \sum a_{ij}e_{j}; \text{ for some }a_{ij} \in A $
We can rewrite this system of equations as : $\begin{align*} (\alpha-a_{11}e_{1}-a_{12}e_{2}-a_{13}e_{3}-\cdots &=0\\ -a_{21}e_{1}+(\alpha - a_{22})e_{2}- a_{23}e_{3}- \cdots&= 0\\ \cdots &=0 \end{align*}$
Let $C$ be the matrix of coefficients on the left-hand side. Then Cramers formula tells us that $\det (C)e_{i}=0$ for all $i$. As $M$ is faithful and the $e_{i}$ generate $M$, this implies that $\det(C)=0$. On expanding out the determinant, we obtain an equation:
$\alpha^{n} + c_{1}\alpha^{n-1} + c_{2}\alpha^{n-2} + \cdots+ c_{n} = 0 ,\qquad c_{i} \in A$
From this proof of the Proposition, can anybody please tell me the proof for the theorem of the lecturer?