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

[Math Amateur]

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

indexed by an arbitrary index set,

the product

is defined by the set of all functions

such that

for all i which becomes an

R-Module under pointwise operations,

and

"

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

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 $$ M_1 = A, M_2 = B, M_3 = C $$

Then we have

$$ \Pi M_i = M_1 \times M_2 \times M_3 = A \times B \times C $$

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

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

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

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

Can someone please help and clarify this matter?
====================================Note: Since the operations in $$ A \times B \times C $$ would, i imagine be as follows:

$$ (a_1, b_1, c_1) + (a_2, b_2, c_2) = (a_1 + a_2, b_1 + b_2, c_1 + c_2) $$

and

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

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

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

Can someone confirm this?

Mind you I am guessing and cannot see why this follows from $$ \alpha, \beta : I \rightarrow \cup \{ M_i | i \in I \}$$ such that $$ \alpha (i) \in M_i $$ for all i

 

[Evgeny.Makarov]

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

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

Then we have

$$ \Pi M_i = M_1 \times M_2 \times M_3 = A \times B \times C $$

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

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

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

So my problem here is what precisely are the functions $$ \alpha , \beta $$ 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 $$ \alpha: I \rightarrow \cup \{ M_i\mid i \in I \}$$ such that $$ \alpha (i) \in M_i $$ for all $i$?

Note: Since the operations in $$ A \times B \times C $$ would, i imagine be as follows:

$$ (a_1, b_1, c_1) + (a_2, b_2, c_2) = (a_1 + a_2, b_1 + b_2, c_1 + c_2) $$

and

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

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

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

Can someone confirm this?

Yes, your understanding of the structure of $\alpha$ and $\beta$ and operations on them is correct.

 

[Math Amateur]

Thank you for your help and guidance, Evgeny

Peter