Make an addition and multiplication table for ring $\Bbb{Z}_{12}$ with ideal $\left \{ 0,3,6,9 \right \}$.

Question

Make an addition and multiplication table for ring $\Bbb{Z}_{12}$ with ideal $\left \{ 0,3,6,9 \right \}$.

I know how to make addition/multiplication tables, but I am confused as to how to find the elements for the quotient ring $\Bbb{Z}_{12}/\left \{ 0,3,6,9 \right \}$. DanZimm's answer mentioned the three elements but not how they arrived at them.

Answer 1

Let $I = \{0,3,6,9\}$. The elements of the quotient $\Bbb{Z}_{12}/I$ are of the form $a + I$ where $a \in \Bbb{Z}_{12}$. There are only $3$ choices:

$$ 0 + I = \{0,3,6,9\} \\ 1 + I = \{1,4,7,10\} \\ 2 + I = \{2, 5, 8, 11\} $$

There are no more distinct elements. Note that $3 + I = I$ and $4 + I = 1+ I$, etc.