How to interpret quotient rings of gaussian integers

[nateHI]

Homework Statement

This is just a small part of a larger question and is quite simple really. It's just that I want to confirm my understanding before moving on.

What are some of the elements of $Z<i>/I</i>$ where I is an ideal generated by a non-zero non-unit integer. For the sake of argument, let's take I=<3>.

Homework Equations

The Attempt at a Solution

Representatives from one coset would be the following...
(8+4i)/<3>=(5+i)/<3>=(2+i)$\in Z<i>/<3></i>$

Representatives from another coset would be the following...
(7+5i)/<3>=(4+2i)/<3>=(1+2i)$\in Z<i>/<3></i>$

 

[micromass]

At first glance I think it would be $\mathbb{Z}_3$. Can you try to prove this?

 

[nateHI]

http://www.math.uconn.edu/~kconrad/blurbs/ugradnumthy/Zinotes.pdf

I found the answer in that link on page 17. You are correct the elements are isomorphic to the integers mod 3 adjoin i. I just didn't understand that the quotient ring I was trying to produce the elements of was the (integers adjoin i) mod the generator of the ideal.