Why are maximal ideals prime?

Could anyone explain to me why maximal ideals are prime?

I'm approaching it like this, let $R$ be a commutative ring with $1$ and let $A$ be a maximal ideal. Let $a,b\in R:ab\in A$.

I'm trying to construct an ideal $B$ such that $A\subset B \neq A$ As this would be a contradiction. An alternative idea I had was to prove that $R/A$ is an integral domain, but this reduces to the same problem.

EDIT: Ergh.. just realized that I've learnt a theorem that states is $A$ is a maximal ideal then $R/A$ is a field

Here’s a proof that doesn’t involve the quotient $R/A$.

Suppose that $A$ is not prime; then there are $a,b\in R\setminus A$ such that $ab\in A$. Let $B$ be the ideal generated by $A \cup \{a\}$; $B = \{x+ar: x\in A\text{ and }r\in R\}$. Clearly $A \subsetneq B$, so $B = R$, $1_R \in B$, and hence $1_R = x + ar$ for some $x\in A$ and $r\in R$. Then $$b = b1_R = b(x+ar) = bx + bar.$$ But $bx \in bA \subseteq RA = A$, and $bar \in Ar \subseteq AR = A$, so $b \in A$. This contradiction shows that $A$ is prime.

$A$ is an maximal ideal $\Rightarrow$ $R/A$ is a field $\Rightarrow$ $R/A$ is an integral domain $\Rightarrow$ $A$ is prime

Let $A$ be a maximal ideal. Then $R/A$ contains no proper ideals, by the correspondence theorem. Indeed, $R/A$ is a field (assuming that $R$ contains an identity). Hence, $A$ is a prime ideal.

Theorem. $R/A$ is a field.

Proof. Let $i+A\in R/A$ such that $i+A\neq 0+A$. We want to prove that $i+A$ is a unit. So set $B=A+Ri=\{a+ri: a\in A, r\in R\}$.

Now, you (yourself!) need to prove that $B$ is an ideal, and that $A\subset B$ properly. Since $A$ is maximal this means that $B=R$.

As $B=R$ we have $1 \in B$, hence there exists some $a\in A, r\in R$ such that $a + ri = 1$. Then $1+A=(a+ri)+A=ri+A=(r+A)(i+A)$, and so $i+A$ is a unit, as required. QED

Worth emphasis: the common proof in Brian's answer is simply an ideal-theoretic form of the common Bezout-based proof of $ $ Euclid's Lemma $ $ for integers. To highlight the analogy we successively translate the Bezout-based proof into the language of gcds and (principal) ideals.

Euclid's Lemma in Bezout form, gcd form and ideal form
${\!\begin{align} Ax\!+\!ay=&\,\color{#c00}1,\,\ A\ \mid\ ab\ \ \ \Rightarrow\, A\ \mid\ b.\ \ \ {\bf Proof}\!:\, A\ \mid\ Ab,ab\, \Rightarrow\, A\,\mid Abx\!\!+\!aby = (\!\overbrace{Ax\!+\!ay}^{\large\color{#c00} 1}\!) b = b\\ (A,\ \ \ a)=&\,\color{#c00}1,\,\ A\ \mid\ ab\ \ \ \Rightarrow\, A\ \mid\ b.\ \ \ {\bf Proof}\!:\, A\ \mid\ Ab,ab\, \Rightarrow\, A\,\mid (Ab,\ \ ab) = (A,\ \ \ a)\ \ b =\, b\\ A\!+\!(a)=&\,\color{#c00}1,\,\ A\supseteq\! (ab)\, \Rightarrow\, A \supseteq\! (b).\, {\bf Proof}\!:\, A \supseteq Ab,ab \,\Rightarrow A\supseteq Ab\!+\!(ab)\! =(A\!+\!(a))b =\! (b)\\ A +{\cal A}\ =&\,\color{#c00}1,\,\ A\supseteq {\cal A B}\, \Rightarrow\, A \supseteq\, {\cal B}.\,\ {\bf Proof}\!:\, A\, \supseteq\! A{\cal B},\!{\cal AB}\!\Rightarrow\! A\supseteq A{\cal B}\!+\!\!{\cal AB} =(A+{\cal A}){\cal B} = {\cal B} \end{align}}$

The third ideal form is precisely the same proof as in Brian's answer. The last form shows that the proof works more generally for coprime (i.e. comaximal) ideals $\ A,\, {\cal A},\ $ i.e. $\ A+{\cal A}= 1. $ In the second proof for integers, we can read $\,(A,a)\,$ either as a gcd or an ideal. Read as a gcd the proof employs the universal property of the gcd $\, d\mid m,n\iff d\mid (m,n)\,$ and the gcd distributive law $\,(Ab,ab) = (A,a)b.\,$ In the first proof (by Bezout) the gcd arithmetic is traded off for integer arithmetic, so the the gcd distributive law becomes the distributive law in the ring of integers.