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I am reading Dummit and Foote, Section 10.4: Tensor Products of Modules. I am currently studying Example 3 on page 369 (see attachment).
Example 3 on page 369 reads as follows: (see attachment)
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In general,
$$ \mathbb{Z} / m \mathbb{Z} \otimes_\mathbb{Z} \mathbb{Z} / n \mathbb{Z} \cong \mathbb{Z} / d \mathbb{Z}$$ where d is the g.c.d. of the integers m and n.
To see this observe first that
$$ a \otimes b = a \otimes (b \cdot 1) = (ab) \otimes 1 = ab(1 \otimes 1) $$
... ... ... etc etc ...
... The map
$$ \phi \ : \ \mathbb{Z} / m \mathbb{Z} \times_\mathbb{Z} \mathbb{Z} / n \mathbb{Z} \to \mathbb{Z} / d \mathbb{Z} $$
defined by
$$ \phi (a mod \ m , b mod \ n ) = ab mod \ d $$
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 $$ \phi $$ 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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Example 3 on page 369 reads as follows: (see attachment)
-------------------------------------------------------------------------------
In general,
$$ \mathbb{Z} / m \mathbb{Z} \otimes_\mathbb{Z} \mathbb{Z} / n \mathbb{Z} \cong \mathbb{Z} / d \mathbb{Z}$$ where d is the g.c.d. of the integers m and n.
To see this observe first that
$$ a \otimes b = a \otimes (b \cdot 1) = (ab) \otimes 1 = ab(1 \otimes 1) $$
... ... ... etc etc ...
... The map
$$ \phi \ : \ \mathbb{Z} / m \mathbb{Z} \times_\mathbb{Z} \mathbb{Z} / n \mathbb{Z} \to \mathbb{Z} / d \mathbb{Z} $$
defined by
$$ \phi (a mod \ m , b mod \ n ) = ab mod \ d $$
is well defined since d divides both m and n. ... ...
... ...
-----------------------------------------------------------------------------
Can someone please help with the following issue:
What is meant by the map $$ \phi $$ 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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