For a prime integer $p$ is $pR$ a maximal ideal in $R$?

Question

If $R$ is a commutative ring with unit and $p$ is a prime number, then is $pR$ a maximal ideal of $R$? If not what conditions should I impose on $R$?

Answer 1

A silly example : Consider the field of real numbers, $\mathbb{R}$. Clearly $p\mathbb{R}=\mathbb{R}$, so $\mathbb{R}/p\mathbb{R} \cong \lbrace 0\rbrace$. More interesting: Let $R=\mathbb{Z}[x]$. $\mathbb{Z}[x]/p\mathbb{Z}[x] \cong (\mathbb{Z}/p\mathbb{Z})[x]$ which is not a field!

Answer 2

There are also plenty of counterexamples apart from fields. For example $p$ could be a unit even if $R$ is not a field, e.g. in $\mathbf Z [\frac{1}{p}]$, or $p$ could fail to be irreducible, e.g in $\mathbf Z[i]$ we have $(1+i)(1−i)=2$.

Answer 3

A prime $p$ is not prime in $\mathbb Z[\sqrt p]$ because $(\sqrt p)^2 \in p\mathbb Z[\sqrt p]$ but $\sqrt p \notin p\mathbb Z[\sqrt p]$. Thus $p\mathbb Z[\sqrt p]$ is not a prime ideal of $\mathbb Z[\sqrt p]$.

Therefore, no prime remains prime in the ring of all algebraic integers.