Why is every ideal in $\mathbb Z[ i ] / (5)$ principal? Let $R = \mathbb Z[ i ] / (5)$ .
Prove that any ideal in $R$ is principal. 
Any ideas on how to prove this?

Your ring is isomorphic to $\Bbb{Z}[x]/(5,x^2+1) \cong \Bbb{F}_5[x]/(x^2 + 1) $. Now notice that $x^2 + 1 = (x+2)(x+3)$ in here. The ideals $(x+2)$, $(x+3)$ are coprime since $x+3 - x-2 = 1$ and hence by the Chinese Remainder Theorem
$$\Bbb{F}_5[x]/(x^2 + 1) \cong \Bbb{F}_5[x]/(x+2) \oplus \Bbb{F}_5[x](x+3) \cong \Bbb{F}_5 \times \Bbb{F}_5.$$
This is a product of PIRs and hence is a PIR. Alternatively, $\Bbb{Z}[i]$ is a Dedeking domain since it is the ring of integers of $\Bbb{Q}(i)$, and hence by this exercise here is a PIR.
