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?

$\mathbb{Z}[i]$ is a Euclidean domain hence it is a principal ring. Consider the homomorphism $\phi:\mathbb{Z}[i]\rightarrow \mathbb{Z}[i]/\left<5\right>$ that sends $x$ to $x+\left<5\right>$. This homomorphism is surjective, thus $\mathbb{Z}[i]/\left<5\right>$ is a principal ring.
