Showing two rings are not isomorphic

  • Thread starter Thread starter samtiro
  • Start date Start date
  • Tags Tags
    Rings
samtiro
Messages
3
Reaction score
0

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?
 
Physics news on Phys.org
Yes, this is correct.

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

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
View full post »

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
View full post »

Similar threads

I Isomorphisms between C4 & Z4 Groups
Replies
1
Views
2K
Why a group is not a direct or semi direct product
Replies
3
Views
2K
Ring homomorphism from Z4 to Z8
Replies
7
Views
7K
Isomorphic Quotient Groups in Z4 x Z4
Replies
7
Views
12K
Show that the rings Z[x] and Z are not isomorphic
Replies
4
Views
2K
Showing that two rings are not isomorphic
Replies
1
Views
2K
Show that two rings are not isomorphic
Replies
3
Views
2K
Showing that an element is a unit in a ring
Replies
3
Views
2K
Showing isomorphism between fractions and a quotient ring
Replies
1
Views
2K
Showing when quadratic integer rings are isomorphic
Replies
7
Views
2K

Hot Threads

Recent Insights

Back
Top