I am trying to make the tables for:
1) $\mathbb{Z}_6/\langle\overline{3}\rangle$
2) $(\mathbb{Z}_2 \times \mathbb{Z}_3)/\langle(\overline{1},\overline{0})\rangle$
I have never seen a table for a quotient ring, so I am unsure on how I should do it. I have a guess that it should be, for multiplication, like $\mathbb{Z}_3$:
\begin{array}{l | c c c } * & \widetilde{0} & \widetilde{1} & \widetilde{2} \\ \hline \widetilde{0} & \widetilde{0} & \widetilde{0} & \widetilde{0} \\ \widetilde{1} & \widetilde{0} & \widetilde{1} & \widetilde{2} \\ \widetilde{2} & \widetilde{0} & \widetilde{2} & \widetilde{1} \end{array}
Because $\mathbb{Z}/\mathbb{2Z}$ have the table as $\mathbb{Z}_2$ because it contains only the remainder $0$ or $1$, for even and odd.
Am I onto something here? If you are to construct the table: could you do it as I can perceive it by the definition of quotient ring as well?
Thank you in advance!
This is a strangely long-winded answer.
To form the multiplication table for $\mathbb Z_6/(3)=(\mathbb Z/6\mathbb Z)/ (3)=(\mathbb Z/6 \mathbb Z)/(3 \mathbb Z/6\mathbb Z) \cong \mathbb Z/3 \mathbb Z$ by the ring isomorphism theorems. We can also see this with our bare hands. In $\mathbb Z/6 \mathbb Z=\{0,1,2,3,4,5\}$, which of these elements are divisible by $3$? These are the ones that are $0$ in the kernel. In particular, the quotient should look like $\{0,1,2\}$ as a set. This is because $5 \equiv 2$ and $4 \equiv 1$ modulo $3$. Anyhow, you can then form the multiplication table exactly like $\mathbb Z_3$.
A note: There is generally a nice way to view $\mathbb Z/6 \mathbb Z$, and basically the trick is to notice that $3 \cdot 2=6$ and $3,2$ are coprime. So, for example, take $(1,1) \in \mathbb Z_3 \times \mathbb Z_2$ and just start adding it to repeatedly (or looking at its multiples): $(1,1),(2,0),(0,1),(1,0),(2,1),(0,0),(1,1)$. Notice that this is an element of order $6$ under addition, and that $(1,1)$ is the multiplicative identity.
Question 1: consider the ring homomorphism $\mathbb Z/6 \mathbb Z \to \mathbb Z/3 \mathbb Z \times \mathbb Z/2 \mathbb Z$ given by $1 \mapsto (1,1)$. What is its kernel? where does $3$ get mapped to? What can you say about the relationship between the two questions?
Question 2: What is $\langle \overline{3}\rangle$ isomorphic to (as a ring) inside of $\mathbb Z/6 \mathbb Z$? Look at its multiples.