Direct Products and Sums of Modules - Notation - 2nd Post

In summary, John Dauns book "Modules and Rings" is difficult to understand because of his notation for products and sums of modules.
  • #1
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I am reading John Dauns book "Modules and Rings". I am having problems understanding the notation of Section 1-2 Direct Products and Sums (pages 5-6) - see attachment).

In section 1-2.1 Dauns writes:

================================================== ======

"1-2.1 For any arbitrary family of modules
png.latex
indexed by an arbitrary index set,

the product
png.latex
is defined by the set of all functions

png.latex
such that
png.latex
for all i which becomes an

R-Module under pointwise operations,
png.latex
and

png.latex
"

================================================== ======

I have tried a simple example in rder to understand Dauns notation.

Consider a family of right R-Modules A, B, and C

Let I be an index set I = {1,2,3} so that \(\displaystyle M_1 = A, M_2 = B, M_3 = C \)

Then we have

\(\displaystyle \Pi M_i = M_1 \times M_2 \times M_3 = A \times B \times C \)

My problem now is to understand (exactly) the set of functions

\(\displaystyle \alpha, \beta : I \rightarrow \cup \{ M_i | i \in I \}\) such that \(\displaystyle \alpha (i) \in M_i \) for all i

where I am assuming that \(\displaystyle \cup \{ M_i | i \in I \} = M_1 \cup M_2 \cup M_3 \)

So my problem here is what precisely are the functions \(\displaystyle \alpha , \beta \) in this example.

Can someone please help and clarify this matter?

====================================Note: Since the operations in \(\displaystyle A \times B \times C \) would, i imagine be as follows:

\(\displaystyle (a_1, b_1, c_1) + (a_2, b_2, c_2) = (a_1 + a_2, b_1 + b_2, c_1 + c_2) \)

and

\(\displaystyle (a_1, b_1, c_1)r = (a_1r, b_1r, c_1r)\) for \(\displaystyle r \in R \)

one would imagine that \(\displaystyle \alpha (1) = a_1 , \alpha (2) = b_1 , \alpha (3) = c_1 \)

and \(\displaystyle \beta (1) = a_2 , \beta (2) = b_2 , \beta (3) = c_2 \)

Can someone confirm this?

Mind you I am guessing and cannot see why this follows from \(\displaystyle \alpha, \beta : I \rightarrow \cup \{ M_i | i \in I \}\) such that \(\displaystyle \alpha (i) \in M_i \) for all i
 
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  • #2
Peter said:
Consider a family of right R-Modules A, B, and C

Let I be an index set I = {1,2,3} so that \(\displaystyle M_1 = A, M_2 = B, M_3 = C \)

Then we have

\(\displaystyle \Pi M_i = M_1 \times M_2 \times M_3 = A \times B \times C \)

My problem now is to understand (exactly) the set of functions

\(\displaystyle \alpha, \beta : I \rightarrow \cup \{ M_i | i \in I \}\) such that \(\displaystyle \alpha (i) \in M_i \) for all i

where I am assuming that \(\displaystyle \cup \{ M_i | i \in I \} = M_1 \cup M_2 \cup M_3 \)

So my problem here is what precisely are the functions \(\displaystyle \alpha , \beta \) in this example.

Can someone please help and clarify this matter?
Well, you described $\alpha$ and $\beta$. There are two ways to look at them, which are isomorphic. The easiest is to see $\alpha$ as an ordered triple $(\alpha_i,\alpha_2,\alpha_3)$ where $\alpha_i\in M_i$ for $i=1,2,3$. Now, if instead of $\alpha_i$ we write $\alpha(i)$, this shows the second way: $\alpha$ is a single function from $\{1,2,3\}$ such that $\alpha(i)\in M_i$ for $i=1,2,3$. Do you see that \(\displaystyle \alpha: I \rightarrow \cup \{ M_i\mid i \in I \}\) such that \(\displaystyle \alpha (i) \in M_i \) for all $i$?

Peter said:
Note: Since the operations in \(\displaystyle A \times B \times C \) would, i imagine be as follows:

\(\displaystyle (a_1, b_1, c_1) + (a_2, b_2, c_2) = (a_1 + a_2, b_1 + b_2, c_1 + c_2) \)

and

\(\displaystyle (a_1, b_1, c_1)r = (a_1r, b_1r, c_1r)\) for \(\displaystyle r \in R \)

one would imagine that \(\displaystyle \alpha (1) = a_1 , \alpha (2) = b_1 , \alpha (3) = c_1 \)

and \(\displaystyle \beta (1) = a_2 , \beta (2) = b_2 , \beta (3) = c_2 \)

Can someone confirm this?
Yes, your understanding of the structure of $\alpha$ and $\beta$ and operations on them is correct.
 
  • #3
Thank you for your help and guidance, Evgeny

Peter
 

FAQ: Direct Products and Sums of Modules - Notation - 2nd Post

1. What is a direct product of modules?

A direct product of modules is a new module formed by combining two or more modules together in a specific way. It is denoted by the symbol ⊕ and is defined as the set of all possible combinations of elements from the individual modules, with operations defined component-wise.

2. How is the direct product of modules different from the direct sum?

The direct product and direct sum of modules are similar in that they both involve combining multiple modules. However, the direct product is a more general form of combination, allowing for more flexibility in the types of modules that can be combined. The direct sum, on the other hand, is a more restrictive form of combination, requiring the modules to have the same underlying structure.

3. What is the notation used for direct products of modules?

The notation for direct products of modules is ⊕, which is read as "direct sum". This symbol is used to represent both the direct product and direct sum, as they are often used interchangeably in mathematics.

4. Can the direct product of modules be written as a tensor product?

Yes, the direct product of modules can be written as a tensor product. This is because both the direct product and tensor product involve combining elements from different modules. However, the two operations have different properties and should not be confused with each other.

5. How does the direct product of modules relate to the direct sum of modules?

The direct product and direct sum of modules are related in that the direct product is a more general form of the direct sum. This means that the direct sum is a special case of the direct product, where the combined modules have the same underlying structure. In essence, the direct sum is a subset of the direct product.

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