Showing two rings are not isomorphic

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In summary, Z4 x Z4 is not isomorphic to Z16 because the number of units in each ring is different, with Z4 x Z4 having 4 units and Z16 having 8 units. This difference in the properties of the rings prevents an isomorphism between them.
  • #1
samtiro
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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?
 
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  • #2
Yes, this is correct.

You could also have said that the rings are not isomorphic, since they are not even isomorphic as groups.
 
  • #3
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!
 

1. How do you determine if two rings are not isomorphic?

Two rings are not isomorphic if they do not have the same number of elements or if their addition and multiplication tables do not match up.

2. Can two rings with the same number of elements be non-isomorphic?

Yes, two rings can have the same number of elements but still be non-isomorphic if their addition and multiplication tables do not match up.

3. Is there a quick way to show that two rings are not isomorphic?

There is no quick way to show that two rings are not isomorphic. It involves checking all possible ways to map one ring to the other and seeing if they preserve the ring structure.

4. Can two rings with different operations be isomorphic?

No, two rings with different operations (e.g. one with addition and multiplication, and the other with addition and exponentiation) cannot be isomorphic because they have different algebraic structures.

5. Is it possible for two rings to be non-isomorphic but have the same underlying set?

Yes, it is possible for two rings to have the same underlying set but be non-isomorphic if their operations (addition and multiplication) do not match up. In this case, the rings are considered to be different structures, even though they have the same elements.

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