MHB Tensor Products - D&F page 369 Example 3 - The map phi

Math Amateur
Gold Member
MHB
Messages
3,920
Reaction score
48
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)

-------------------------------------------------------------------------------

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

----------------------------------------------------------------------------
 
Physics news on Phys.org
This is pretty basic.

When one defines a mapping on a quotient object (which is typically induced by some equivalence relation) one has to make sure the mapping depends only on the equivalence classes themselves, and not "representative elements".

For example, the map:

$f:\Bbb Z_5 \to \Bbb Z_4$ "defined" by $f(a\text{ (mod }5)) = a\text{ (mod }4)$

is not well-defined, because:

$[3]_5 = [8]_5$ but $f([3]_5) = [3]_4$ whereas $f([8]_5) = [0]_4$.

So to prove that $\phi$ is well-defined, we need to show that if:

$a \equiv a'\text{ (mod }m)$
$b \equiv b'\text{ (mod }n)$

that:

$ab \equiv a'b'\text{ (mod }d)$.

Now the first condition means that:

$a - a' = km$ for some $k \in \Bbb Z$, and the second condition means that:
$b - b' = sn$ for some $s \in \Bbb Z$.

Thus:

$ab - a'b' = ab - a'b + a'b - a'b' = (a - a')b + a'(b - b') = kmb + a'sn$.

Since $d|m$ and $d|n$, we have:

$m = td$
$n = ud$ for some (positive) integers $t,u$.

Hence:

$ab - a'b' = kmb + a'sn = k(td)b + a's(ud) = (ktb + a'su)d$

which shows that:

$ab \equiv a'b'\text{ (mod }d)$.
 
Peter said:
... 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. ... ...

This is how the map is being define $\varphi: \mathbb{Z}_m \times \mathbb{Z}_n \to \mathbb{Z}_d$, for $(\alpha,\beta) \in \varphi: \mathbb{Z}_m \times \mathbb{Z}_n \to \mathbb{Z}_d$ we choose representatives $\alpha = [a]_n$ and $\beta = _m$. In other words, $\alpha,\beta$ are classes mod $n$ and $m$ and can be represented by some integer. Then we define $\alpha \beta\in \mathbb{Z}_d$ to be the class mod $d$ that is represented by the integer $ab$. The question here is how do we know that if we picked different representatives in the beginning that we end up in the same class in $\mathbb{Z}_d$?

The next step is to check that this map $\varphi: \mathbb{Z}_m \times \mathbb{Z}_n \to \mathbb{Z}_d$ is bilinear. From here it will follow from the universal property of tensor product that there is a morphism $\psi : \mathbb{Z}_m \otimes \mathbb{Z}_n \to \mathbb{Z}_d $ which is defined by $\psi (a\otimes b) = ab$.
 
I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
Back
Top