- Thread starter
- #1

- Jun 22, 2012

- 2,918

Example 3 on page 369 reads as follows: (see attachment)

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In general,

\(\displaystyle \mathbb{Z} / m \mathbb{Z} \otimes_\mathbb{Z} \mathbb{Z} / n \mathbb{Z} \cong \mathbb{Z} / d \mathbb{Z}\) where d is the g.c.d. of the integers m and n.

To see this observe first that

\(\displaystyle a \otimes b = a \otimes (b \cdot 1) = (ab) \otimes 1 = ab(1 \otimes 1) \)

... ... ... etc etc ...

... The map

\(\displaystyle \phi \ : \ \mathbb{Z} / m \mathbb{Z} \times_\mathbb{Z} \mathbb{Z} / n \mathbb{Z} \to \mathbb{Z} / d \mathbb{Z} \)

defined by

\(\displaystyle \phi (a mod \ m , b mod \ n ) = ab mod \ d \)

is well defined since d divides both m and n. ... ...

... ...

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Can someone please help with the following issue:

What is meant by the map \(\displaystyle \phi \) being 'well defined' and why is d dividing both m and n important in this matter?

I would appreciate some help.

Peter

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