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Thread: Direct Products and Sums of Modules - Notation

  1. #1
    Super Member Bernhard's Avatar
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    Direct Products and Sums of Modules - Notation

    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:

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

    "1-2.1 For any arbitrary family of modules $\displaystyle M_i, i \in I $ indexed by an arbitrary index set, the product $\displaystyle \Pi \{ M_i | i \in I \} \equiv \Pi M_i $ is defined by the set of all functions $\displaystyle \alpha , \beta : I \rightarrow \cup \{ M_i | i \in I \} $ such that $\displaystyle \alpha (i) \in M_i $ for all i which becomes an R-Module under pointwise operations, $\displaystyle ( \alpha - \beta) (i) = \alpha (i) - \beta (i) $ and $\displaystyle ( \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 $\displaystyle M_1 \times M_2 \times M_3 $ is not neccesarily the same as $\displaystyle M_2 \times M_1 \times M_3 $ and so on.

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

    Then the product $\displaystyle \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 $\displaystyle \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 $\displaystyle M_ 1, M_2, M_3 $ the operations in $\displaystyle M_1 \times M_2 \times M_3 $ would be of the form

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

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

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

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

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

    Can anyone clarify these matters for me?

    Peter
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    Last edited by Bernhard; Aug 9th 2013 at 06:17 PM.
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    Re: Direct Products and Sums of Modules - Notation

    Hi Bernhard,
    I hope the attached discussion helps.

    Direct Products and Sums of Modules - Notation-mhfrings6.png
    Thanks from Bernhard
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    Super Member Bernhard's Avatar
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    Re: Direct Products and Sums of Modules - Notation

    Thank you johng

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

    My problem is that the elements of these products are ordered triples not just sets - so the elements of $\displaystyle A \times B \times C $ are of the form $\displaystyle (a_i, b_i, c_i) $ while the ordered triples of $\displaystyle B \times A \times C $ are of the form $\displaystyle (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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