Welcome to our community

Be a part of something great, join today!

Direct Products and Sums of Modules - Notation

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 in section 1-2 (see attachment)

My issue is understanding the notation on Section 1-2, subsection 1-2.1 (see attachment).

Dauns is dealing with the product [TEX] \Pi \{ M_i | i \in I \} \equiv \Pi M_i [/TEX] and states in (ii) - see attachment

------------------------------------------------------------------------------------------------------
Alternatively, the product can be viewed as consisting of all strings or sets

[TEX] x = \{ x_i | i \in I \} \equiv (x_i)_{i \in I} \equiv (x_i) \equiv ( \_ \_ \_ \ , x_i , \_ \_ \_ ) , x_i \in M_i; [/TEX] i-th

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

I am not sure of the meaning of the above set of equivalences. Can someone briefly elaborate ... preferably with a simple example

If we take the case of I = {1,2,3} and consider the product [TEX] M_1 \times M_2 \times M_3 [/TEX] then does Dauns notation mean

[TEX] x = (x_1, x_2, x_3) [/TEX] where order in the triple matters (mind you if it does what are we to make of the statement [TEX] x = \{ x_i | i \in I \} [/TEX]


Can someone confirm that [TEX] x = (x_1, x_2, x_3) [/TEX] is a correct interpretation of Dauns notation?

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

Dauns then goes on to define the direct sum as follows:

The direct sum [TEX] \oplus \{ M_i | i \in I \} \equiv \oplus M_i [/TEX] is defined as the submodule [TEX] \oplus M_i \subseteq \Pi M_i [/TEX] consisting of those elements [TEX] x = (x_i) \in \Pi M_i [/TEX] having at most a finite number of non-zero coordinates or components. Sometimes [TEX] \oplus M_i , \Pi M_i [/TEX] are called the external direct sum and the external direct product respectively.


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

Can someone point out the difference between [TEX] \oplus M_i , \Pi M_i [/TEX] in the case of the example involving [TEX] M_1, M_2, M_3 [/TEX] - I cannot really see the difference! For example, what elements exactly are in [TEX] \Pi M_i [/TEX] that are not in [TEX] \oplus M_i [/TEX]

I would be grateful if someone can clarify these issues.

Peter

This has also been posted on MHF
 

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,492
If we take the case of I = {1,2,3} and consider the product [TEX] M_1 \times M_2 \times M_3 [/TEX] then does Dauns notation mean

[TEX] x = (x_1, x_2, x_3) [/TEX] where order in the triple matters (mind you if it does what are we to make of the statement [TEX] x = \{ x_i | i \in I \} [/TEX]
Yes, if $x\in M_1 \times M_2 \times M_3$, then $x$ is an ordered triple $(x_1,x_2,x_3)$ such that $x_i\in M_i$ for $i=1,2,3$. The notation $x = \{ x_i\mid i \in I \}$ is perhaps unfortunate. It would be valid if $x_i$ carried the information of the module they came from, e.g., $x = \{ (x_i,i)\mid i \in I \}$. Then it is possible to order such set according to the second component of its elements.

Alternatively, $x$ may be viewed as a function from $\{1,2,3\}$ to $M_1\cup M_2\cup M_3$ with the restriction that $x(i)$ is always in the correct module $M_i$. Such construction is called dependent product in programming (type theory).

Can someone point out the difference between [TEX] \oplus M_i , \Pi M_i [/TEX] in the case of the example involving [TEX] M_1, M_2, M_3 [/TEX] - I cannot really see the difference! For example, what elements exactly are in [TEX] \Pi M_i [/TEX] that are not in [TEX] \oplus M_i [/TEX]
For finite products and sums (i.e., when the index set is finite), direct product and direct sum are exactly the same. See Wikipedia.
 

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
Thanks Evgeny ... A most helpful post ...

Peter