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February 14th, 2018, 06:23 AM  #1 
Member Joined: Sep 2011 Posts: 99 Thanks: 1  Isomorphism problem
I have gotten the following answer to (a) and (b) which require verification on them. I have also attached the theorem for reference. (a) Z x Z => have zero divisors The matrix has no zero divisors (no nonzero matrix when multiplied to the matrix gives zero element) Hence not isomorphic. (b) Z x Z => have 2 elements Z x Z subscript 5 => have 5 elements ( [0,0] [0,1] [0,2] [0,3] [0,4] ) Hence not isomorphic. 
February 14th, 2018, 07:55 AM  #2 
Senior Member Joined: Oct 2009 Posts: 751 Thanks: 253 
(a) You should prove this matrix ring has no zero divisors (b) $\mathbb{Z}\times \mathbb{Z}_5$ has a lot more than 5 elements. 
February 14th, 2018, 06:38 PM  #3 
Member Joined: Jan 2016 From: Athens, OH Posts: 92 Thanks: 47 
1. Actually the matrix $$\begin{bmatrix}a&b\\0&0\end{bmatrix}$$ does have zero divisors. You show this. For the proper answer, ask yourself if all elements of the matrix ring commute. 2. Does the matrix ring have a multiplicative identity? 
February 15th, 2018, 06:09 AM  #4 
Member Joined: Sep 2011 Posts: 99 Thanks: 1 
I have done up part a as attached and may need verification on that. Next for part b, I am not sure how I should go about doing. May need help on that. Thanks 
February 15th, 2018, 06:44 AM  #5 
Member Joined: Jan 2016 From: Athens, OH Posts: 92 Thanks: 47 
Your image is hard for me to read, but I think you correctly showed that the matrix ring is not commutative. Here's some hints for the second question: 
February 16th, 2018, 06:21 AM  #6 
Member Joined: Sep 2011 Posts: 99 Thanks: 1 
thanks all for the help.


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