# Find all elements of quotient ring

I am studying the definitions of rings, ideals, and quotient ring, but I have a bit problem to apply the theory into the practice.

I would like to find all elements of quotient ring $\mathbb{Z}[i]/I$ , where $\mathbb{Z}[i] =\{ {a+bi|a, b∈ \mathbb{Z}}\}$ - Gaussian integers and $I$ is ideal $I = (2 + 2i)$.

How one can find all elements of such a quotient ring? What is the algorithm?

In this case, we are beginning with the Gaussian integers -- which have the form $a+bi$ for integers $a$ and $b$ -- and introducing the identity $2+2i=0$. What that relation does is allow you to replace any occurrence of $2i$ with $-2$. (Or vice versa). The effect of this is that you can rewrite any Gaussian integer in the form $a + bi$ where $b$ is either 0 or 1 -- just keep peeling off multiples of $2i$ and exchanging them for $-2$. For example, in this ring $$4 + 7i = 4 + (1 + 6)i = 4 + i + 6i = 4 - 6 + i = -2 + i$$
• Some elements are of the form $a$, where $a \in \mathbb{Z}$, and those elements add and multiply just like integers normally do.
• The other elements are of the form $a + i$, where $a \in \mathbb{Z}$, and those elements combine according to the law $(a+i) + (b+i) = a + b - 2$.
Here $\mathbb{Z}[i]/(2+2i)=\{z\in\mathbb{Z}[i]\,|2+2i=0\}=\{\pm1,\pm i,\pm1\pm i,\pm 1 \mp i\}$, we basically consider $\mathbb{Z}[i]$ under modulo $2+2i$.