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Homework Help: Abstract alg zero divisor?

  1. Nov 5, 2005 #1
    Let R be the set of all real valued functions defined for all reals under function addition and multiplication.

    Determine all zero divisors of R.

    A zero divisor is a non zero element such that when multiplied with another nonzero element the product is zero.
    So I said that the zero divisors of R would be all the functions in R that are not the zero function but take on the value of zero at least one time.
    Am I close?
    acoording to my answer though f=sin X and g= x-2 would be zero divisors because niether function is the zero function but fg = (sin X)(x-2) = 0 at x=2
    but my teacher said this is wrong he said the product has to be zero for all x.
    How would I even begin to find all the functions that are not the zero function but when multiplied together is zero for all x?
     
    Last edited: Nov 5, 2005
  2. jcsd
  3. Nov 5, 2005 #2

    HallsofIvy

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    How are you defining multiplication in this case? Are you saying that if a function is 0 for any x then it is the "zero" function?
     
  4. Nov 5, 2005 #3
    The problem says under normal function multiplication. The zero function is
    Z(x)= 0 for all x
     
  5. Nov 5, 2005 #4

    HallsofIvy

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    So f(x)= 2x is 0 for x= 0. But f*g is not, in general, equal to the 0 function.
     
  6. Nov 5, 2005 #5
    so are there any zero divisors
     
  7. Nov 6, 2005 #6

    Hurkyl

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    One thing I would do on a problem like this if I got stuck is to try and specialize the definitions to the situation at hand.

    You know that f is a zero divisor iff there exists g such that f*g = 0.

    So if f, g, and 0 are all functions, what form does this condition take?
     
  8. Nov 6, 2005 #7

    HallsofIvy

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    What about, for example, f(x)= 0 if x<0, 1 if x>= 0; g(x)= 1 if x< 0; 0 if x>=0?
     
  9. Nov 6, 2005 #8
    the piecewise functions above are both zero divisors. since neither function is the zero function but thier product is zero for all x, ok I think I get now.

    so any function that is not the zero function , but takes on the value of zero for some x, is a zero divisor. Becasue if my function f takes on zero for some interval I, I can always construct another function g that is zero on I compliment.

    just to make sure I get this, f(x) = x^2 + 1 cannot be zero divisor because I would have to multiply it by a function g thats always zero, which would make g the zero function.
     
    Last edited: Nov 6, 2005
  10. Nov 6, 2005 #9

    HallsofIvy

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    It doesn't have to be "intervals". Let f(x)= 0 if x is rational, 1 if x is irrational, g(x)= 1 if x is rational, 0 if x is irrational.

    Your second statement is correct: if f(x) is NEVER 0 then it is not a zero divisor.
     
  11. Nov 6, 2005 #10

    Hurkyl

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    Well, technically, if a function is zero anywhere, then it is zero on some interval (plus some other points).
     
  12. Nov 6, 2005 #11

    HallsofIvy

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    Do you mean you are considering a singleton {a} as an interval?
     
  13. Nov 6, 2005 #12

    Hurkyl

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    Sure, since {a} = [a, a]
     
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