Finding all ideals N of $Z_{12}$ and compute $Z_{12}/N$ in each case.

Question

I already find all the ideals N of $Z_{12}$.

Here's what I've got:

Since ideals must be additive subgroups, by group theory we see that the possibilities are restricted to the cyclic additive subgroups,

$<0> = {0}$

$<1> = {0,1,2,3,4,5,6,7,8,9,10,11}$

$<2> = {0,2,4,6,8,10}$

$<3> = {0,3,6,9}$

$<4> = {0,4,8}$

And

$<6> = {0,6}$

Now, my problem is how can I compute for $Z_{12}/N$ in each case; that is, find a known ring to which the quotient ring is isomorphic.

Thank You in advance for the help!

Answer 1

You might find the third isomorphism theorem helpful here if you think of $\mathbb{Z}_{12}\cong \mathbb{Z}/12\mathbb{Z}$.

Answer 2

If you look at it a bit more carefully you'll find every quotient rings of yours being isomorphic to the subrings you have mentioned above using the fact that if the ring is cyclic with respect to the first operation then every subring is so. And here $Z_{12}$ is a cyclic group