Addition and Multiplication table for Ring/Ideal

Question

I'm not sure if it's possible to show it here, but how would the

addition and multiplication table look like for R/I (where R is rings with ideal I) when $$ R = Z_{12} \text{ and } I = \{0,3,6,9\} $$

Answer 1

We can see that there will be $3$ elements in $R / I$, they are: $$ \widetilde{0} = 0 + I\\ \widetilde{1} = 1 + I \\ \widetilde{2} = 2 + I $$ for any other element you can form can be broken down first $\mod{12}$ and then using the ideal $I$. Now we can construct our addition and multiplication tables: $$ \begin{array}{l | c c c } * & \widetilde{0} & \widetilde{1} & \widetilde{2} \\ \hline \widetilde{0} & \widetilde{0} & \widetilde{0} & \widetilde{0} \\ \widetilde{1} & \widetilde{0} & \widetilde{1} & \widetilde{2} \\ \widetilde{2} & \widetilde{0} & \widetilde{2} & \widetilde{1} \end{array} \quad \begin{array}{l | c c c } + & \widetilde{0} & \widetilde{1} & \widetilde{2} \\ \hline \widetilde{0} & \widetilde{0} & \widetilde{1} & \widetilde{2} \\ \widetilde{1} & \widetilde{1} & \widetilde{2} & \widetilde{0} \\ \widetilde{2} & \widetilde{2} & \widetilde{0} & \widetilde{1} \end{array} $$

Hopefully this helps!