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I am reading Dummit and Foote Section 10.5 on Exact Sequences.
I am trying to understand Example 1 as given at the bottom of page 381 and continued at the top of page 382 - please see attachment for the diagram and explanantion of the example.
The example, as you can no doubt see, requires an understanding of the nature of the quotient module (\mathbb{Z} / m \mathbb{Z} ) / (n \mathbb{Z} / m \mathbb{Z} )
To make this quotient more tangible, in this example take m = 6, n = 3 so k = 2.
Then we are trying to understand the nature of the quotient module (\mathbb{Z} / 6 \mathbb{Z} ) / (3 \mathbb{Z} / 6 \mathbb{Z} )
Now consider the nature of (\mathbb{Z} / 6 \mathbb{Z} )
We have 0 + \mathbb{Z} / 6 \mathbb{Z} = { ... ... -18, -12, -6, 0 , 6, 12, 18, 24, ... ... }
and 1 + \mathbb{Z} / 6 \mathbb{Z} = {... ... -17, -11, -5, 1, 7, 13, 19, 25, ... }
and so on
But what is 3 \mathbb{Z} / 6 \mathbb{Z} ? and indeed, further, what is (\mathbb{Z} / 6 \mathbb{Z} ) / (3 \mathbb{Z} / 6 \mathbb{Z} ) ?
Can someone please help clarify this matter?
Peter
I am trying to understand Example 1 as given at the bottom of page 381 and continued at the top of page 382 - please see attachment for the diagram and explanantion of the example.
The example, as you can no doubt see, requires an understanding of the nature of the quotient module (\mathbb{Z} / m \mathbb{Z} ) / (n \mathbb{Z} / m \mathbb{Z} )
To make this quotient more tangible, in this example take m = 6, n = 3 so k = 2.
Then we are trying to understand the nature of the quotient module (\mathbb{Z} / 6 \mathbb{Z} ) / (3 \mathbb{Z} / 6 \mathbb{Z} )
Now consider the nature of (\mathbb{Z} / 6 \mathbb{Z} )
We have 0 + \mathbb{Z} / 6 \mathbb{Z} = { ... ... -18, -12, -6, 0 , 6, 12, 18, 24, ... ... }
and 1 + \mathbb{Z} / 6 \mathbb{Z} = {... ... -17, -11, -5, 1, 7, 13, 19, 25, ... }
and so on
But what is 3 \mathbb{Z} / 6 \mathbb{Z} ? and indeed, further, what is (\mathbb{Z} / 6 \mathbb{Z} ) / (3 \mathbb{Z} / 6 \mathbb{Z} ) ?
Can someone please help clarify this matter?
Peter