existence of maximal ideals


Let R be a unital ring. Every proper idealMathworldPlanetmathPlanetmath of R lies in a maximal idealMathworldPlanetmath of R.

Applying this theorem to the zero idealMathworldPlanetmathPlanetmath gives the following corollary:


Every unital ring R0 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 be a proper ideal of , and let Σ be the partially ordered setMathworldPlanetmath

Σ={𝒜𝒜 is an ideal of , and 𝒜}

ordered by inclusion.

Note that 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 {𝒜α} be a non-empty chain of ideals in Σ, so for all indices α,β we have

𝒜α𝒜β or 𝒜β𝒜α.

We claim that defined by


is a suitable upper bound.

  • is an ideal. Indeed, let a,b, so there exist α,β such that a𝒜α, b𝒜β. Since these two ideals are in a totally orderedPlanetmathPlanetmath chain we have

    𝒜α𝒜β or 𝒜β𝒜α

    Without loss of generality, we assume 𝒜α𝒜β. Then both a,b𝒜β, and 𝒜β is an ideal of the ring . Thus a+b𝒜β.

    Similarly, let r and b. As above, there exists β such that b𝒜β. Since 𝒜β is an ideal we have




    Therefore, is an ideal.

  • , otherwise 1 would belong to , so there would be an α such that 1𝒜α so 𝒜α=. But this is impossible because we assumed 𝒜αΣ for all indices α.

  • . Indeed, the chain is non-empty, so there is some 𝒜α in the chain, and we have 𝒜α.

Therefore Σ. Hence every chain in Σ has an upper bound in Σ and we can apply Zorn’s Lemma to deduce the existence of , a maximal elementMathworldPlanetmath (with respect to inclusion) in Σ. By definition of the set Σ, this must be a maximal ideal of containing . 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 ChoiceMathworldPlanetmath (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