Intersection of maximal ideal with subring

[coquelicot]

Let R be an integral ring (eventually can be supposed integrally closed), and R' an integral extension of R. Assume that M is a maximal ideal of R'. Must the intersection of M with R be a maximal ideal of R ?

Thx.

 

[Erland]

Is an integral ring the same as an integral domain? And what's is an integral extension? (I get the impression that you mean somrthing like the extension of the natural numbers to the integers.)

 

[micromass]

Let R be an integral ring (eventually can be supposed integrally closed), and R' an integral extension of R. Assume that M is a maximal ideal of R'. Must the intersection of M with R be a maximal ideal of R ?

Thx.

This follows from the following two lemma's:

Lemma 1: If $R\subseteq R^\prime$ are rings such that $R^\prime$ is integral over $R$, and if $J$ is an ideal of $R^\prime$ and if $I=R\cap J$, then $R^\prime/J$ is integral over $R/I$.

Proof: Take $x\in R^\prime$, then we have some equation
$$x^n + a_1x^{n-1} + ... + a_n = 0$$
with $a_i \in R$. Reducing this modulo $J$ then yields that $x+J$ is integral over $R/I$.

Lemma 2: Let $R\subseteq R^\prime$ be integral domains such that $R^\prime$ is integral over $R$. Then $R$ is a field if and only if $R^\prime$ is a field.

Proof: If $R$ is a field, then let $y\in R^\prime$ be nonzero. Then there is some equation
$$y^n + a_1y^{n-1} + ... + a_n = 0$$
with $a_i \in R$. We can take this equation of smallest possible degree. But then
$$y^{-1} = -a_n^{-1}(y^{n-1} + a_1 y^{n-2} + ... + a_{n-1})$$
note that $a_n\neq 0$ because of the integral domain requirement. Thus $R^\prime$ is a field.

Conversely, if $R^\prime$ is a field and if $x\in R$ is nonzero, then $x^{-1}$ is integral over $R$ and thus satisfies an equation
$$x^{-n} + a_1x^{-n+1} + ... + a_n=0$$
It follows that $x^{-1} = - (a_1 + a_2x + ... + a_nx^{n-1})\in R$.

 

[coquelicot]

FOR ERLAND : 1) Yes, I meant integral domain. 2) I meant that every element x of R' is integral over R.
MICROMASS : You are the best of the best. thanks a lot.

 

[mathwonk]

If you want a reference for this topic, the so called "going up theorem", see page 3 of Mumford's red book:

https://www.amazon.com/dp/354063293X/?tag=pfamazon01-20