Prime ideal and maximal ideal of an integral domain

Question

In an integral domain, $\{0\}$ is always a prime ideal. What about maximal ideal? $(0)$ is a prime ideal in $\mathbb{Z}$ which is an I.D but it is not maximal in $\mathbb{Z}$. So I can not conclude that $(0)$ is always maximal ideal in I.D, right? Then what about an example of ideal which is prime in $\mathbb{R}$ ( real numbers) but not maximal? As every field is an integral domain, can I conclude that $(0)$ is not always maximal ideal of field?

Answer 1

Let $R$ be a commutative ring with $1$. You can see in many books related to the topic ( Atiyah-Macdonald, 1.2 and if you want more expression, Lemma 3.1 of the book Steps in Commutative Algebra by Sharp ):

$R$ is a field if and only if $R$ has exactly two ideals: $0$ and $R$ itself.

Case 1. $R$ is a field. (e.g $\mathbb{R}$)
In this case $0$ is the only proper ideal (Hence maximal ideal) of $R$.

Case 2. The domain $R$ is not a field.
In this case $R$ has an ideal other than $0$, say $I$. Since $0\subsetneq I$, you know that $0$ is not maximal.

Answer 2

Hint: what are the ideals in a field? More generally, do you know the fact that for any commutative ring $R$ and ideal $I$, $R/I$ is a field if and only if $I$ is maximal?