Direct Products and Sums of Modules - Notation

**Bernhard** · August 9th 2013, 06:09 PM

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:

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"1-2.1 For any arbitrary family of modules $M_i, i \in I$ indexed by an arbitrary index set, the product $\Pi \{ M_i | i \in I \} \equiv \Pi M_i$ is defined by the set of all functions $\alpha , \beta : I \rightarrow \cup \{ M_i | i \in I \}$ such that $\alpha (i) \in M_i$ for all i which becomes an R-Module under pointwise operations, $( \alpha - \beta) (i) = \alpha (i) - \beta (i)$ and $( \alpha (i) r for r \in R$ "

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

Can anyone clarify this notation for me - preferably using a simple example.

One specific issue is the following:

I imagine (possibly incorrectly) that the order or sequence of modules matters - that is $M_1 \times M_2 \times M_3$ is not neccesarily the same as $M_2 \times M_1 \times M_3$ and so on.

However, if we take $I = \{ 1, 2, 3\}$ then $\{ M_i | i \in I \} = \Pi \{M_1, M_2, M_3 \}$

Then the product $\Pi \{ M_i | i \in I \} \equiv \Pi \{ M_1, M_2, M_3 \}$

BUT the set notation implies that the order or sequence of the product does not matter i,e that $\Pi \{ M_1, M_2, M_3 \} = M_1 \times M_2 \times M_3 = M_3 \times M_1 \times M_2$ etc

Is this correct?

It worries me that, again considering the example of $M_ 1, M_2, M_3$ the operations in $M_1 \times M_2 \times M_3$ would be of the form

$(x_1, x_2, x_3 ) + (y_1, y_2, y_3) = (x_1 + y_1, x_2 + y_2, x_3 + y_3)$

$(x_1, x_2, x_3 ) a = (x_1a, x_2a, x_3a )$

and in these triples (certainly in things like curves in 3 space) the order, I think, would matter.

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Then, I have to say I do not really have a good understanding of or feeling for what follows regarding the functions $\alpha$ and $\beta$ . In particular, in the pointwise operations why choose subtraction instead of addition.

Can anyone clarify these matters for me?

Peter

**johng** · August 9th 2013, 08:27 PM

Hi Bernhard,
I hope the attached discussion helps.

Direct Products and Sums of Modules - Notation-mhfrings6.png

**Bernhard** · August 9th 2013, 09:30 PM

Thank you johng

I am still struggling with the idea that, in your example $A \times B \times C$ is the same as (isomorphic to) $B \Times A \times C$ .

My problem is that the elements of these products are ordered triples not just sets - so the elements of $A \times B \times C$ are of the form $(a_i, b_i, c_i)$ while the ordered triples of $B \times A \times C$ are of the form $(b_i, a_i, c_i)$ . As ordered triples these are not the same (of course, they are the same as sets of components - but I think the product of modules is dealing with elements that are ordered triples?

Can you comment.

I am also still struggling with the product as a set of mappings. Can you give some examples of the mappings in the case of A, B and C?

Peter

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