Showing two rings are not isomorphic

[samtiro]

Homework Statement

Explain why Z4 x Z4 is not isomorphic to Z16.

Homework Equations

Going to talk about units in a ring.
Units are properties preserved by isomorphism.

The Attempt at a Solution

We see the only units in Z4 are 1 and 3.
So the units of Z4 x Z4 are (1,1) , (3,3) , (1,3) , (3,1)

The Units of Z16 are 1,3,5,7,9,11,13,15.

So there are 4 units in Z4 x Z4 but in Z16 we have 8 units. So there can not be an isomorphism between the two.

Is this correct?

 

[micromass]

Yes, this is correct.

You could also have said that the rings are not isomorphic, since they are not even isomorphic as groups.

 

[samtiro]

oh ok. We actually are doing rings before groups so I would not have been able to use groups that is why i resorted to using units.

Thanks for confirming my answer!