existence of maximal ideals

existence of maximal ideals

Theorem

.

Let

R

be a

unital ring

.
Every

proper ideal

of

R

lies in a

maximal ideal

of

R

.

Applying this

theorem

to the

zero ideal

gives the following corollary:

Corollary

.

Every unital ring

R

≠

0

has a maximal ideal.

Proof of theorem

.
This proof is a straightforward application of Zorn’s Lemma.
Readers are encouraged to attempt the proof themselves
before reading the details below.

Let

ℐ

be a proper ideal of

ℛ

,
and let

Σ

be the

partially ordered set

Σ

=

{

𝒜

∣

𝒜

⁢

is an ideal of

⁢

ℛ

,

and

⁢

ℐ

⊆

𝒜

≠

ℛ

}

ordered by

inclusion

.

Note that

ℐ

∈

Σ

, so

Σ

is non-empty.

In order to apply Zorn’s Lemma
we need to prove that every non-empty chain (

http://planetmath.org/TotalOrder

)
in

Σ

has an

upper bound

in

Σ

.
Let

{

𝒜

α

}

be a non-empty chain of ideals in

Σ

,
so for all indices

α

,

β

we have

𝒜

α

⊆

𝒜

β

or

𝒜

β

⊆

𝒜

α

.

We claim that

ℬ

defined by

ℬ

=

⋃

α

𝒜

α

is a suitable upper bound.

•

ℬ

is an ideal. Indeed, let

a

,

b

∈

ℬ

,
so there exist

α

,

β

such that

a

∈

𝒜

α

,

b

∈

𝒜

β

. Since these two
ideals are in a

totally ordered

chain we have

𝒜

α

⊆

𝒜

β

or

𝒜

β

⊆

𝒜

α

Without loss of generality, we assume

𝒜

α

⊆

𝒜

β

. Then both

a

,

b

∈

𝒜

β

, and

𝒜

β

is an
ideal of the ring

ℛ

. Thus

a

+

b

∈

𝒜

β

⊆

ℬ

.

Similarly, let

r

∈

ℛ

and

b

∈

ℬ

. As
above, there exists

β

such that

b

∈

𝒜

β

.
Since

𝒜

β

is an ideal we have

r

⋅

b

∈

𝒜

β

⊆

ℬ

and

b

⋅

r

∈

𝒜

β

⊆

ℬ

.

Therefore,

ℬ

is an ideal.

•

ℬ

≠

ℛ

, otherwise

1

would belong to

ℬ

, so there would be an

α

such that

1

∈

𝒜

α

so

𝒜

α

=

ℛ

. But
this is impossible because we assumed

𝒜

α

∈

Σ

for all indices

α

.

•

ℐ

⊆

ℬ

.
Indeed, the chain is non-empty,
so there is some

𝒜

α

in the chain,
and we have

ℐ

⊆

𝒜

α

⊆

ℬ

.

Therefore

ℬ

∈

Σ

. Hence every chain in

Σ

has an upper bound in

Σ

and we can apply Zorn’s Lemma to
deduce the

existence

of

ℳ

, a

maximal element

(with
respect to inclusion) in

Σ

.
By definition of the set

Σ

,
this

ℳ

must be a maximal ideal of

ℛ

containing

ℐ

. QED

Note that the above proof never makes use of
the associativity of ring multiplication,
and the result therefore holds also in non-associative rings.
The result cannot, however, be generalized to rings without unity.

Note also that the use of the

Axiom of Choice

(in the form of Zorn’s Lemma)
is necessary,
as there are models of ZF in which the above theorem and corollary fail.

Title

existence of maximal ideals

Canonical name

ExistenceOfMaximalIdeals

Date of creation

2013-03-22 13:56:57

Last modified on

2013-03-22 13:56:57

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

22

Author

yark (2760)

Entry type

Theorem

Classification

msc 16D25

Classification

msc 13A15

Synonym

existence of maximal ideals

Related topic

ZornsLemma

Related topic

AxiomOfChoice

Related topic

MaximalIdeal

Related topic

ExistenceOfMaximalSubgroups

Related topic

DefinitionOfPrimeIdealByKrull

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