Finding the element of the quotient ring $Z[i]/\langle 2+2i\rangle$

Question

First, I'm writing an element to confirm whether I understood this quotient ring correctly.

$$(5 + 7i) + \langle 2+2i \rangle = 2(2+2i) + (1+3i) + \langle 2+2i \rangle = (1+3i) + \langle 2+2i \rangle$$

Is this correct?

Does $4$ belong in the given ideal $\langle 2+2i \rangle$? If yes, how can I see it?

Answer 1

Your first calculation looks fine to me.

For the second question, you need to decide if $4=c\cdot(2+2i)$ for some $c\in \mathbb{Z}[i]$. Solve the equation and see what you get.