(adsbygoogle = window.adsbygoogle || []).push({}); 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?

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# Quick way to tell if two rings are isomorphic?

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