General question about quotient rings

Question

I hope to better understand the notion of a quotient ring through this example:

I am given $R=\mathbb{Z}[i]=\{a+bi:a,b\in \mathbb{Z}\}$ and $M=\{a+bi: 3|a,3|b\}$. I have already shown that $M$ is a maximal ideal of $R$, but I am also asked to show that $R/M$ is a field with $9$ elements. I understand that $R/M$ is the set of cosets of $M$ in $R$, but for rings I am confused as to what these look like. For instance, do I look at $k+M$ or $kM$ for $k\in R$? How am I to show that there are $9$ cosets?

$\mathbf{Note:}$ If you can clarify what $R/M$ looks like for me, I think I would be able to find the answer on my own.

Answer 1

Hint: You could write out the multiplication table, but there's an easier argument: What happens when you mod out by a maximal ideal?

Quotient rings are made up of additive cosets, i.e. the elements of your quotient ring $R/M$ look like $k+M$ with $k\in R$.

Here are a few cosets, all distinct from eachother: $M$, $1+M$, $1+i+M$. I also claim that $2+M=5+3i+M=2+6i+M$ (why?). Hopefully this is enough for you to see what's going on. Feel free to comment if you still have questions.

Answer 2

$M$ is $(3)$, the principal ideal generated by $3$ in $\mathbb{Z}[i]$. Can you think of a good set of representatives for all the cosets of $(3)$? What values of $a$ and $b$ might you pick for $(a + bi) + (3)$?

Answer 3

This used to confuse me as well. Generally, the line of thinking that helps me is that two cosets $k$ and $k^{'}$ are equivalent mod $M$ if and only if $k-k^{'}$ is an element of $M$. In your example that means two cosets $\overline{a+bi}$, $\overline{c+di}$ are equivalent if and only if $3 \mid a-c$ and $3 \mid b-d$. Think about which kinds of elements satisfy this. You should be able to think of 9 natural coset representatives.