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.
**Hints:**  

**|.** Try to show that every non-zero ideal of $\mathbb Z$ is generated by an element in the ideal with minimal absolute value.  

**||.** Show that an ideal $(n)$ is prime iff $n$ is prime iff the ideal is maximal.  

Hope this helps.