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- #1

- Jun 22, 2012

- 2,937

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 [TEX] (\mathbb{Z} / m \mathbb{Z} ) / (n \mathbb{Z} / m \mathbb{Z} ) [/TEX]

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 [TEX] (\mathbb{Z} / 6 \mathbb{Z} ) / (3 \mathbb{Z} / 6 \mathbb{Z} ) [/TEX]

Now consider the nature of [TEX] (\mathbb{Z} / 6 \mathbb{Z} ) [/TEX]

We have [TEX] 0 + \mathbb{Z} / 6 \mathbb{Z} [/TEX] = { ... ... -18, -12, -6, 0 , 6, 12, 18, 24, ... ... }

and [TEX] 1 + \mathbb{Z} / 6 \mathbb{Z} [/TEX] = {... ... -17, -11, -5, 1, 7, 13, 19, 25, ... }

and so on

But what is [TEX] 3 \mathbb{Z} / 6 \mathbb{Z} [/TEX] ? and indeed, further, what is [TEX] (\mathbb{Z} / 6 \mathbb{Z} ) / (3 \mathbb{Z} / 6 \mathbb{Z} ) [/TEX] ?

Can someone please help clarify this matter?

Peter