A maximal ideal (in say a commutative ring ) is an ideal which is maximal among proper ideals. (This is a second-order definition, as it quantifies over subsets of .)
Equivalently, an ideal is maximal if the quotient ring is a field. This suggests a first-order definition: an ideal is maximal if .
(every proper ideal is contained in a maximal one)
Assuming the axiom of choice then:
Let be a commutative ring and let be a proper ideal. Then contains a maximal ideal containing , i.e. .
(See also at prime ideal theorem.)
Write for the set of proper ideals of , partially ordered by inclusion. We claim that every chain in 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 is a chain. We have to produce an such that for all then .
We claim that such is provided by the union:
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 is an ideal, consider . There are thus with and . Since a chain is total order by definition, either or . We may assume the former without restriction, otherwise rename . Therefore now and so we may add them there and find that . Similarly if then .
Finally to see that this idea is indeed proper. But since all the are proper, neither of them contains , and hence does not contain .
Maximal ideals in the spectrum of a commutative ring correspond precisely to the closed points in the Zariski topology on (this prop.).
Closed points are at the heart of the definition of schemes. A scheme 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 ( a field) factors through one of the embeddings.
Last revised on August 21, 2024 at 02:40:16. See the history of this page for a list of all contributions to it.