Question: In the ring $\mathbb{Z}$ the following conditions on a nonzero ideal $I$ are equivalent: (i) $I$ is prime; (ii) $I$ is maximal; (iii) $I=(p)$ with $p$ prime.

Answer: 
HINT: $\mathbb{Z}/I$ is a field if and only if $I$ is a maximal ideal.
**HINT:** $\mathbb{Z}$ is a Principal Ideal Domain (PID).