Integrality of an element of the quotient field of a domain Let $R$ be a Noetherian domain with quotient field $K$,
$I \subseteq R$ a finitely generated ideal, $I \neq (0)$ and
$x \in K$ such that $x \cdot I \subseteq I$.
I want to show that $x$ is integral over $R$.
Supposedly, this is also true when dropping the requirement of $R$ being Noetherian.
My ideas so far: 


*

*Let $x = \frac a b$ with $a,b \in R$. If we could show that $\frac 1 b$ is integral over $R$, then also $\frac a b$ is integral over $R$.

*Using something similar as https://math.stackexchange.com/a/371657/514331 :
We have $\frac 1 b \cdot I \subseteq \frac 1 {b^2} \cdot I \subseteq \dots$
If we could show that  $\frac 1 b \cdot I \subseteq I$, then by $R$ being Noetherian, we would get $\frac 1 {b^n} \cdot I = \frac 1 {b^{n+1}} \cdot I$ for some $n$. But even with this, I fail to see that $\frac 1 b$ is integral over $R$.

 A: Here is a proof which uses Noetherianness (not that it is much simpler). You have inclusions of $R$-algebras, $R\subset R[x]\subset \mathrm{End}_R(I)$. Noetherian property implies the last is a finite type $R$-module and hence again by Noetherian property, so is $R[x]$. This immediately implies $x$ is integral over $R$. 
A: Your statement is not quite correct, you need to assume that $I$ is also nonzero. In that case however, there is a very elementary way to see this.
So you have $I=(f_1,\ldots,f_n)$ and $x\cdot  I\subseteq I$. In particular,
$$
x\cdot f_i = \sum_{j=1}^n x_{ij} f_j
$$
for a matrix $X:=(x_{ij})\in R^{n\times n}$. Denote by $f=(f_1,\ldots,f_n)\in R^n$ the vector containing the $f_i$, then we can write this as $X\cdot f = x\cdot f$. Let $\chi=\sum_{k=0}^n a_k t^k \in R[t]$ be the characteristic polynomial of $X$, then we have
$$
\chi(x)\cdot f= \sum_{k=0}^n a_k x^k\cdot f = \sum_{k=0}^n a_k X^k \cdot f = \chi(X)\cdot f = 0
$$
Since $I$ is not the zero ideal, we may assume that $f_1\ne 0$ and the above implies $\chi(x)\cdot f_1 = 0$, which as an equation inside the field $K$ implies $\chi(x)=0$. Now $\chi$ is the integral relation for $x$ which you seek.
PS: Do you have any interesting examples of such a situation?
