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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 $\displaystyle \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

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

$\displaystyle 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 $\displaystyle x = \{ x_i | i \in I \} $

Can someone confirm that $\displaystyle 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 $\displaystyle \oplus \{ M_i | i \in I \} \equiv \oplus M_i $ is defined as the submodule $\displaystyle \oplus M_i \subseteq \Pi M_i $ consisting of those elements $\displaystyle x = (x_i) \in \Pi M_i $ having at most a finite number of non-zero coordinates or components. Sometimes $\displaystyle \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 $\displaystyle \oplus M_i , \Pi M_i $ in the case of the example involving $\displaystyle M_1, M_2, M_3 $ - I cannot really see the difference! For example, what elements exactly are in $\displaystyle \Pi M_i $ that are not in $\displaystyle \oplus M_i $

I would be grateful if someone can clarify these issues.

Peter

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.

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 $\displaystyle x = (x_1, x_2, x_3) $ is a correct interpretation of Dauns notation?"

Peter

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

Hi Peter,

See the attachment.

Attachment 28970

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

Thanks Johng ... Most helpful post!

Peter