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.

Is maximal ideal always prime?

In a commutative ring with unity, every maximal ideal is a prime ideal. The converse is not always true: for example, in any nonfield integral domain the zero ideal is a prime ideal which is not maximal.

Why is every maximal ideal prime?

Since J is a maximal ideal of A, A/J is a field. Because every field is an integral domain, A/J is an integral domain. Since A/J is an integral domain, J is a prime ideal. Thus, every maximal ideal J of A is a prime ideal.