# Quick way to tell if two rings are isomorphic?

by Silversonic
Tags: isomorphic, rings
 P: 130 1. The problem statement, all variables and given/known data Say if the following rings isomorphic to $\mathbb{Z}_6$ (no justification needed); 1) $\mathbb {Z}_2 \times \mathbb {Z}_3$ 2) $\mathbb {Z}_6 \times \mathbb {Z}_6$ 3) $\mathbb {Z}_{18} / [(0,0) , (2,0)]$ 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), $\mathbb {Z}_6 \times \mathbb {Z}_6$ has 36 elements, whereas $\mathbb {Z}_6$ has 6. But then, how do I show the other two? My notes say the first one is isomorphic to $\mathbb {Z}_6$ 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?
Emeritus