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Quick way to tell if two rings are isomorphic? 
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#1
May1512, 03:48 PM

P: 130

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? 


#2
May1512, 03:52 PM

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P: 4,500

The first one you can do using the Chinese Remainder Theorem if you've seen it. If not,
Your third one doesn't seem to make any sense though, since Z_{18} doesn't contain any elements called (0,0) or (2,0) in any standard notation that I have seen 


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