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Math Help - Sum and Product of Functions

  1. #1
    Super Member Bernhard's Avatar
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    Unhappy Sum and Product of Functions

    On PlanetMath.Org regarding the Sum and Product of Functions we find the following:

    Let A be a set and K a field or skew field. If f: A  \rightarrow K and g:A  \rightarrow K, then one can define the product of functions f and g as the function fg: A  \rightarrow K as follows:

    (fg)(x) := f(x) . g(x) for all x belonging to A


    My question is: Why cannot K be a group or a ring with . being the product concerned. Why does K have to be a field or a skew field?

    Bernhard
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Sum and Product of Functions

    Quote Originally Posted by Bernhard View Post
    My question is: Why cannot K be a group or a ring with . being the product concerned. Why does K have to be a field or a skew field?
    More general, we only need a binary operation * :K\times K\to K i.e. for all k,s\in K, k* s\in K . So we can define f*g:A\to K in the way (f* g)(x):=f(x)* g(x) .
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