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Math Help - Direct Products and Sums of Modules - Notation - 2nd Post

  1. #1
    Super Member Bernhard's Avatar
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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  \Pi \{ M_i | i \in I \} \equiv \Pi M_i and states in (ii) - see attachement

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

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

    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?

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

    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.


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

    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
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    Last edited by Bernhard; August 9th 2013 at 07:41 PM.
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    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.
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  3. #3
    Super Member Bernhard's Avatar
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    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
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    Re: Direct Products and Sums of Modules - Notation - 2nd Post

    Hi Peter,
    See the attachment.
    Direct Products and Sums of Modules - Notation - 2nd Post-algebra1.png
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    Super Member Bernhard's Avatar
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    Re: Direct Products and Sums of Modules - Notation - 2nd Post

    Thanks Johng ... Most helpful post!

    Peter
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