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

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
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 \(\displaystyle M_1 = A, M_2 = B, M_3 = C \)

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
 
Last edited:

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,502
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 $i$?

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
Jun 22, 2012
2,918
Thank you for your help and guidance, Evgeny

Peter