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

• Aug 9th 2013, 07:57 PM
Bernhard
Direct Products and Sums of Modules - Notation - 2nd Post
This post follows my previous post on John Dauns book "Modules and Rings"

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

Dauns is dealing with the product $\Pi \{ M_i | i \in I \} \equiv \Pi M_i$ and states in (ii) - see attachement

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

$x = \{ x_i | i \in I \} \equiv (x_i)_{i \in I} \equiv (x_i) \equiv ( \_ \_ \_ \ , x_i , \_ \_ \_ ) , x_i \in M_i;$ i-th

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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 $M_1 \times M_2 \times M_3$ then does Dauns notation mean

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

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

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

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

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

I would be grateful if someone can clarify these issues.

Peter
• Aug 9th 2013, 09:33 PM
johng
Re: Direct Products and Sums of Modules - Notation - 2nd Post
Hi Bernhard,
If the index set I is finite, then certainly the direct product and direct sum are exactly the same. Only for I infinite is the direct sum a proper subset of the direct product.
• Aug 9th 2013, 10:51 PM
Bernhard
Re: Direct Products and Sums of Modules - Notation - 2nd Post
Thanks johng

Are you implicitly answering the following question in the affirmative?

"Can someone confirm that $x = (x_1, x_2, x_3)$ is a correct interpretation of Dauns notation?"

Peter
• Aug 10th 2013, 07:42 AM
johng
Re: Direct Products and Sums of Modules - Notation - 2nd Post
Hi Peter,
See the attachment.
Attachment 28970
• Aug 10th 2013, 04:23 PM
Bernhard
Re: Direct Products and Sums of Modules - Notation - 2nd Post
Thanks Johng ... Most helpful post!

Peter