As with commutative rings, maximal ideals are prime, and also prime ideals contain minimal prime ideals. A ring is a prime ring if and only if the zero ideal is a prime ideal, and moreover a ring is a domain if and only if the zero ideal is a completely prime ideal.

Are prime ideals always maximal?

(1) An ideal P in A is prime if and only if A/P is an integral domain. (2) An ideal m in A is maximal if and only if A/ m is a field. Of course it follows from this that every maximal ideal is prime but not every prime ideal is maximal.

When every prime ideal is maximal ideal?

Every Prime Ideal is Maximal if an=a for any Element a in the Commutative Ring Let R be a commutative ring with identity 1≠0.