Addition and Multiplication table for Ring/Ideal
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!
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paula000
Updated on March 15, 2022Comments

paula000 10 months
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 $$ R = Z_{12} \text{ and } I = \{0,3,6,9\} $$

paula000 over 8 yearsI understand when you said there's 3 elements in R/I, but how did you get the elements after? Where you had 0=0+1...

DanZimm over 8 yearsWell by definition of the quotient ring we have elements of the form $a + I \in R/I$ where $a \in R$. I noticed that $0,1,2$ are each representatives of the three elements in this quotient ring so I name them to be the three distinct elements.

DanZimm over 8 years@paula000 I suppose my answer might not be explanatory enough. Can you try to explain where your confusion arises when you see how I name the three elements in $R/I$? Are you confused on the definition of $R/I$, the definition of an element in $R/I$ or something else altogether?

paula000 over 8 yearsAh okay. I've briefly forgotten the definition of the quotient ring. But yes, I believe understand everything else you explained after. Thank you!

DanZimm over 8 yearsGlad to help, if you get confused later on comment again and I'll try to help out!

Admin over 4 years@DanZimm Hello, may I ask where you got those three elements?