Quick way to tell if two rings are isomorphic?

Silversonic

1. The problem statement, all variables and given/known data

Say if the following rings isomorphic to [itex] \mathbb{Z}_6 [/itex] (no justification needed);

1) [itex] \mathbb {Z}_2 \times \mathbb {Z}_3 [/itex]

2) [itex] \mathbb {Z}_6 \times \mathbb {Z}_6 [/itex]

3) [itex] \mathbb {Z}_{18} / [(0,0) , (2,0)] [/itex]


3. The attempt at a solution

I know how to tell if two rings AREN'T isomorphic - find an essential property that one ring has that another one doesn't. For example part 2), [itex] \mathbb {Z}_6 \times \mathbb {Z}_6 [/itex] has 36 elements, whereas [itex] \mathbb {Z}_6 [/itex] has 6.

But then, how do I show the other two?


My notes say the first one is isomorphic to [itex] \mathbb {Z}_6 [/itex] because 2 and 3 are coprime. But how can it justify that so quickly? Because they are coprime both rings have the same characteristic (i.e. n.1 = 0 for n ≥ 1 in both rings). But is that enough to conclude that they are isomorphic?

And for the last one, where can I begin?


If I can show both rings are generated by their 1, and both rings have the same characteristic, they should be isomorphic right?

Office_Shredder

The first one you can do using the Chinese Remainder Theorem if you've seen it. If not,

This is a good way to do it. In particular if something is isomorphic to Z₆ it should be really easy to construct the isomorphism and prove that it's an isomorphism

Your third one doesn't seem to make any sense though, since Z₁₈ doesn't contain any elements called (0,0) or (2,0) in any standard notation that I have seen