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Math Help - Ring based on families of integers - The Function Ring

  1. #1
    Super Member Bernhard's Avatar
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    Ring based on families of integers - The Function Ring

    I am trying to get a full understanding of Cohn Example 4 on page 9 - see attachment.

    A formal and rigorous example showing clearly the nature and role of the families (and their indicies) involved would help enormously.

    Cohn, page 9, Example 4 states:


    "Let I be any set and denote by  Z^I the set of all functions from I to Z.

    Thus the elements of  Z^I are families  (x_i) of integers indexed by I.

    Addition and multiplication are defined componentwise:

     (x_i) = (y_i) = (z_i) where  z_i = x_i + y_i

    and  (x_i) (y_i) = (t_i) where  t_i = x_i y_i

    It is very easily verified becomes a ring."



    In order to understand this example fully, I would very much appreciate an example of what is at work here showing clearly the nature of the families and their indices.

    Peter
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  2. #2
    Senior Member jakncoke's Avatar
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    Re: Ring based on families of integers - The Function Ring

    Basically it fundamentally comes to the fact that if I is the set of all functions from I \to \mathbb{Z} then I should be countable. Namely.
    there exists a map from  I \to \mathbb{N} .

    adding 2 functions becomes adding their evaluation point wise. \bar{f} + f = \bar{f}(i_n) + f(i_n) for all  \mathbb{N} .

    basically we can take the element in I we attached a label "1" to and evaluate it there for both 2 functions, label "2" and evaluate it there,... for all natural numbers.

    Again, we can do multiplication pointwise as well.
    Thanks from Bernhard
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