Find all elements of quotient ring

Question

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?

Thanks in advance.

Answer 1

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 $2+2i$.

Answer 2

When you mod out by an ideal, you are essentially introducing an identity that allows you to rewrite some elements in a different form.

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 $$4 + 7i = 4 + (1 + 6)i = 4 + i + 6i = 4 - 6 + i = -2 + i$$

So you can think of this ring as having two types of elements in it:

• 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 $(a+i) + (b+i) = a + b - 2$.

To finish this up you should think about what happens when you combine elements of both types. To make sure you really understand what's going on you should also work out what the multiplication laws are for the various types of combinations.