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February 14th, 2018, 07:23 AM   #1
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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
File Type: jpg 5.jpg (13.7 KB, 2 views)
File Type: jpg fo.jpg (18.6 KB, 5 views)
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February 14th, 2018, 08:55 AM   #2
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(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.
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February 14th, 2018, 07:38 PM   #3
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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?
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February 15th, 2018, 07:09 AM   #4
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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
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File Type: jpg Webp.net-resizeimage.jpg (78.1 KB, 4 views)
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February 15th, 2018, 07:44 AM   #5
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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:

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February 16th, 2018, 07:21 AM   #6
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thanks all for the help.
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