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

#### Peter

##### Well-known member
MHB Site Helper
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
$M_i, i \in I$
indexed by an arbitrary index set,

the product
$\Pi \{ M_i | i \in I \} \equiv \Pi M_i$
is defined by the set of all functions

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

R-Module under pointwise operations,
$( \alpha - \beta) (i) = \alpha (i) - \beta (i)$
and

$( \alpha (i) r for r \in R$
"

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

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.

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

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

Last edited:

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
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.

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 $i$?

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.

#### Peter

##### Well-known member
MHB Site Helper
Thank you for your help and guidance, Evgeny

Peter