Example

(1) Let $A$ be a commutative ring. If $m$ is a maximal ideal of $A$, $m$ is prime; the ring $A/m$ is a field (no. 1, Corollary 1 to Theorem 1).

(2) If $A$ is an integral domain, the ideal $\{0\}$ of $A$ is prime (but not maximal in general, as the example of the ring $\mathbf{Z}$ proves).