My Math Forum Isomorphism problem

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February 14th, 2018, 06:23 AM   #1
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Joined: Sep 2011

Posts: 97
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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.
Attached Images
 5.jpg (13.7 KB, 2 views) fo.jpg (18.6 KB, 5 views)

 February 14th, 2018, 07:55 AM #2 Senior Member   Joined: Oct 2009 Posts: 496 Thanks: 164 (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: 89 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
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Joined: Sep 2011

Posts: 97
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
Attached Images
 Webp.net-resizeimage.jpg (78.1 KB, 4 views)

 February 15th, 2018, 06:44 AM #5 Member   Joined: Jan 2016 From: Athens, OH Posts: 89 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: 97 Thanks: 1 thanks all for the help.

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