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

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

Now consider the nature of

We have = { ... ... -18, -12, -6, 0 , 6, 12, 18, 24, ... ... }

and = {... ... -17, -11, -5, 1, 7, 13, 19, 25, ... }

and so on

But what is ? and indeed, further, what is ?

Can someone please help clarify this matter?

Peter