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maximal ideal

Contents

Definition

A maximal ideal (in say a commutative ring RR) is an ideal MM which is maximal among proper ideals. (This is a second-order definition, as it quantifies over subsets of RR.)

Equivalently, an ideal MRM \subseteq R is maximal if the quotient ring R/MR/M is a field. This suggests a first-order definition: an ideal MM is maximal if ( x)¬(xM) yxy=1(\forall_x)\; \neg (x \in M) \Rightarrow \exists_y x y = 1.

Properties

General

Proposition

(every proper ideal is contained in a maximal one)

Assuming the axiom of choice then:

Let RR be a commutative ring and let IRI \subset R be a proper ideal. Then RR contains a maximal ideal 𝔪\mathfrak{m} containing II, i.e. I𝔪I \subset \mathfrak{m}.

(See also at prime ideal theorem.)

Proof

Write PropIdl(R) PropIdl(R)_{\subset} for the set of proper ideals of RR, partially ordered by inclusion. We claim that every chain in PropIdl(R) PropIdl(R)_{\subset} has an upper bound (def.). This then implies the statement by Zorn's lemma (equivalent to the axiom of choice).

To show the claim, assume that 𝒞PropIdl(R) \mathcal{C} \subset PropIdl(R)_{\subset} is a chain. We have to produce an IPropIdl(R)I \in PropIdl(R) such that for all cCc \in C then cIc \subset I.

We claim that such II is provided by the union:

IJ𝒞J. I \coloneqq \underset{J \in \mathcal{C}}{\cup} J \,.

It is clear that if this is indeed a proper ideal, then it is an upper bound of the chain.

To see first of all that this II is an ideal, consider x 1,x 2Ix_1, x_2 \in I. There are thus J 1,J 2𝒞J_1, J_2 \in \mathcal{C} with x 1J 1x_1 \in J_1 and x 2J 2x_2 \in J_2. Since a chain is total order by definition, either J 1J 2J_1 \subset J_2 or J 2J 1J_2 \subset J_1. We may assume the former without restriction, otherwise rename 121 \leftrightarrow 2. Therefore now x 1,x 2J 2x_1, x_2 \in J_2 and so we may add them there and find that x 1+x 2J 2Ix_1 + x_2 \in J_2 \subset I. Similarly if rRr \in R then rx iJ 2Ir x_i \in J_2 \subset I.

Finally to see that this idea II is indeed proper. But since all the J iJ_i are proper, neither of them contains 1R1 \in R, and hence II does not contain 1R1 \in R.

Proposition

In classical mathematics then:

Every maximal ideal is a prime ideal.

Relation to points in the spectrum

Assuming AC and EM, then

Maximal ideals in the spectrum of a commutative ring Spec(R)Spec(R) correspond precisely to the closed points in the Zariski topology on Spec(R)Spec(R) (this prop.).

Closed points are at the heart of the definition of schemes. A scheme XX is a sheaf with respect to the Zariski topology that admits a covering by open embeddings of affine schemes, where “covering” means that every closed point p:Spec(F)Xp: Spec(F) \to X (FF a field) factors through one of the embeddings.

References

Revised on April 25, 2017 11:07:41 by Urs Schreiber (131.220.184.222)