# Quick way to tell if two rings are isomorphic?

## Homework Statement

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

## 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?

Office_Shredder
Staff Emeritus