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 and states in (ii) - see attachement

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

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 then does Dauns notation mean

where order in the triple matters (mind you if it does what are we to make of the statement

Can someone confirm that 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 is defined as the submodule consisting of those elements having at most a finite number of non-zero coordinates or components. Sometimes are called the external direct sum and the external direct product respectively.

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Can someone point out the difference between in the case of the example involving - I cannot really see the difference! For example, what elements exactly are in that are not in

I would be grateful if someone can clarify these issues.

Peter