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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?

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  • $\begingroup$ So 4 is in I and we have the relation 2i = -2 . Therefore there are only 9 elements in this quotient ring: 0,i, -i, 1+i, 1-i, 2+i, 2-i, 3+i, 3-i ; Is this correct? $\endgroup$ – madeel Sep 24 '14 at 7:31
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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.

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