TITLE: If every prime ideal is maximal, what can we say about the ring?
QUESTION [23 upvotes]: Suppose $R$ is a ring and every prime ideal of $R$ is also a maximal ideal of $R$. Then what can we say about the ring $R$?

REPLY [7 votes]: For a commutative ring $R$, the following are equivalent:

Every prime ideal is maximal.
$\dim(R)=0$
$R/\sqrt{0}$ is von Neumann regular
For all $x$ there is some $n \in \mathbb{N}$ such that $x^{n+1}$ divides $x^n$.

There is a big theory about $0$-dimensional commutative rings, see for example David F. Anderson, David Dobbs, "Zero-Dimensional Commutative Rings" (Lecture Notes in Pure and Applied Mathematics).