existence of maximal ideals
Theorem.
Let $\mathrm{R}$ be a unital ring. Every proper ideal^{} of $\mathrm{R}$ lies in a maximal ideal^{} of $\mathrm{R}$.
Applying this theorem to the zero ideal^{} gives the following corollary:
Corollary.
Every unital ring $\mathrm{R}\mathrm{\ne}\mathrm{0}$ 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 $\mathcal{I}$ be a proper ideal of $\mathcal{R}$, and let $\mathrm{\Sigma}$ be the partially ordered set^{}
$$\mathrm{\Sigma}=\{\mathcal{A}\mid \mathcal{A}\text{is an ideal of}\mathcal{R},\text{and}\mathcal{I}\subseteq \mathcal{A}\ne \mathcal{R}\}$$ 
ordered by inclusion.
Note that $\mathcal{I}\in \mathrm{\Sigma}$, so $\mathrm{\Sigma}$ is nonempty.
In order to apply Zorn’s Lemma we need to prove that every nonempty chain (http://planetmath.org/TotalOrder) in $\mathrm{\Sigma}$ has an upper bound in $\mathrm{\Sigma}$. Let $\{{\mathcal{A}}_{\alpha}\}$ be a nonempty chain of ideals in $\mathrm{\Sigma}$, so for all indices $\alpha ,\beta $ we have
$${\mathcal{A}}_{\alpha}\subseteq {\mathcal{A}}_{\beta}\mathit{\hspace{1em}}\text{or}\mathit{\hspace{1em}}{\mathcal{A}}_{\beta}\subseteq {\mathcal{A}}_{\alpha}.$$ 
We claim that $\mathcal{B}$ defined by
$$\mathcal{B}=\bigcup _{\alpha}{\mathcal{A}}_{\alpha}$$ 
is a suitable upper bound.

•
$\mathcal{B}$ is an ideal. Indeed, let $a,b\in \mathcal{B}$, so there exist $\alpha ,\beta $ such that $a\in {\mathcal{A}}_{\alpha}$, $b\in {\mathcal{A}}_{\beta}$. Since these two ideals are in a totally ordered^{} chain we have
$${\mathcal{A}}_{\alpha}\subseteq {\mathcal{A}}_{\beta}\mathit{\hspace{1em}}\text{or}\mathit{\hspace{1em}}{\mathcal{A}}_{\beta}\subseteq {\mathcal{A}}_{\alpha}$$ Without loss of generality, we assume ${\mathcal{A}}_{\alpha}\subseteq {\mathcal{A}}_{\beta}$. Then both $a,b\in {\mathcal{A}}_{\beta}$, and ${\mathcal{A}}_{\beta}$ is an ideal of the ring $\mathcal{R}$. Thus $a+b\in {\mathcal{A}}_{\beta}\subseteq \mathcal{B}$.
Similarly, let $r\in \mathcal{R}$ and $b\in \mathcal{B}$. As above, there exists $\beta $ such that $b\in {\mathcal{A}}_{\beta}$. Since ${\mathcal{A}}_{\beta}$ is an ideal we have
$$r\cdot b\in {\mathcal{A}}_{\beta}\subseteq \mathcal{B}$$ and
$$b\cdot r\in {\mathcal{A}}_{\beta}\subseteq \mathcal{B}.$$ Therefore, $\mathcal{B}$ is an ideal.

•
$\mathcal{B}\ne \mathcal{R}$, otherwise $1$ would belong to $\mathcal{B}$, so there would be an $\alpha $ such that $1\in {\mathcal{A}}_{\alpha}$ so ${\mathcal{A}}_{\alpha}=\mathcal{R}$. But this is impossible because we assumed ${\mathcal{A}}_{\alpha}\in \mathrm{\Sigma}$ for all indices $\alpha $.

•
$\mathcal{I}\subseteq \mathcal{B}$. Indeed, the chain is nonempty, so there is some ${\mathcal{A}}_{\alpha}$ in the chain, and we have $\mathcal{I}\subseteq {\mathcal{A}}_{\alpha}\subseteq \mathcal{B}$.
Therefore $\mathcal{B}\in \mathrm{\Sigma}$. Hence every chain in $\mathrm{\Sigma}$ has an upper bound in $\mathrm{\Sigma}$ and we can apply Zorn’s Lemma to deduce the existence of $\mathcal{M}$, a maximal element^{} (with respect to inclusion) in $\mathrm{\Sigma}$. By definition of the set $\mathrm{\Sigma}$, this $\mathcal{M}$ must be a maximal ideal of $\mathcal{R}$ containing $\mathcal{I}$. QED
Note that the above proof never makes use of the associativity of ring multiplication, and the result therefore holds also in nonassociative 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  20130322 13:56:57 
Last modified on  20130322 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 