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Math Help - Commutative ring functions

  1. #1
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    Commutative ring functions

    Could somebody please solve this problem?
    Let the commutative ring F(R) be the set of all functions R-R equipped with the operations of pointwise addition and pointwise multiplication: given f, g are members F(R), define functions f+g and fg by
    f+g: a→ f(a) + g(a) and fg: →f(a)g(a)
    (Notice fg is not their composite)
    Pointwise addition and multiplication are the operations on functions occurring in calculus for example.
    f(x) g(x)dx =∫f(x) dx +∫g(x) dx
    And D(fg)=D(f)g+f(D(g), where D denotes derivative.
    The sum f+g in the first integrand is pointwise addition while the product fg in the derivative formula is pointwise multiplication.

    Show that the above commutative ring F(R) contains elements f≠0,1 with f2=f
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mwanahisab View Post
    Could somebody please solve this problem?
    Let the commutative ring F(R) be the set of all functions R-R equipped with the operations of pointwise addition and pointwise multiplication: given f, g are members F(R), define functions f+g and fg by
    f+g: a→ f(a) + g(a) and fg: →f(a)g(a)
    (Notice fg is not their composite)
    Pointwise addition and multiplication are the operations on functions occurring in calculus for example.
    f(x) g(x)dx =∫f(x) dx +∫g(x) dx
    And D(fg)=D(f)g+f(D(g), where D denotes derivative.
    The sum f+g in the first integrand is pointwise addition while the product fg in the derivative formula is pointwise multiplication.

    Show that the above commutative ring F(R) contains elements f≠0,1 with f2=f
    How about f(x)=\begin{cases}0\quad\text{if}\quad x=0\\1\quad\text{if}\quad x\ne 0\end{cases}?
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