Show that $R/I$ is isomorphic with $\mathbb{Z}/5\mathbb{Z} \oplus \mathbb{Z}/5\mathbb{Z}$

Question

$R=\{a+ib:\ a,b\in\mathbb{Z}\}$

$I=\{a+ib:\ 5\mid a\text{ and }5\mid b\}$

Ok so I wanted to create homomorphism from $R$ onto $\mathbb{Z}/5\mathbb{Z} \oplus \mathbb{Z}/5\mathbb{Z}$.

$\phi(a+ib)=([a],[b])$ and while I think this is highly suggestive I can't show that $\phi((a+ib)(c+id))=\phi(a+ib)\phi(c+id)$

Should I try another approach and if not , any hint about how to prove equality above would be appreciated.

Answer 1

Hint. Try $\phi(a+ib)=(a-2b,a+2b)$.