Intersection of maximal ideal with subring

In summary, we have discussed the concept of integral rings and extensions, and the relationship between maximal ideals of integral rings and their extensions. We have also examined two lemmas regarding integral domains and fields. The "going up theorem" is a recommended reference for this topic.
  • #1
coquelicot
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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.
 
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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.)
 
  • #3
coquelicot said:
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
[tex]x^n + a_1x^{n-1} + ... + a_n = 0[/tex]
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
[tex]y^n + a_1y^{n-1} + ... + a_n = 0[/tex]
with ##a_i \in R##. We can take this equation of smallest possible degree. But then
[tex]y^{-1} = -a_n^{-1}(y^{n-1} + a_1 y^{n-2} + ... + a_{n-1})[/tex]
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
[tex]x^{-n} + a_1x^{-n+1} + ... + a_n=0[/tex]
It follows that ##x^{-1} = - (a_1 + a_2x + ... + a_nx^{n-1})\in R##.
 
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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.
 
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Related to Intersection of maximal ideal with subring

What is the definition of "intersection of maximal ideal with subring"?

The intersection of a maximal ideal with a subring is the set of all elements that are contained in both the maximal ideal and the subring.

Why is the intersection of maximal ideal with subring important in mathematics?

This intersection helps to identify elements that are common to both the maximal ideal and the subring, and can provide insight into the structure and properties of the ring.

How is the intersection of maximal ideal with subring related to other concepts in algebra?

The intersection of a maximal ideal with a subring is related to other concepts such as subrings, ideals, and quotient rings, as it involves the combination of these concepts in a specific way.

Is the intersection of maximal ideal with subring always a maximal ideal?

No, the intersection of a maximal ideal with a subring may not always be a maximal ideal. It can be a prime ideal, but it depends on the specific ring and subring in question.

Can the intersection of maximal ideal with subring be empty?

Yes, it is possible for the intersection of a maximal ideal with a subring to be empty if there are no elements that are common to both the maximal ideal and the subring.

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