When a prime ideal is a maximal ideal In a commutative ring with unit every maximal ideal is prime. Under what conditions does the converse occur?
When the ring has Krull dimension equal to zero.  If we're talking about integral domains then every prime ideal of $R$ is maximal if and only if $R$ is a field (since $0$ is a prime ideal in any integral domain).
A ring $R$ with identity is artinian if and only if it is noetherian and every prime ideal is a maximal ideal by Hopkins theorem.