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Math Help - rings

  1. #1
    MHF Contributor chiph588@'s Avatar
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    rings

    For which of the following rings is it possible for the product of two nonzero elements to be zero?
    (A) The ring of complex numbers
    (B) The ring of integers modulo 11
    (C) The ring of continuous real-valued functions on [0, 1]
    (D) The ring {  a+b \sqrt{2} :a and b are rational numbers}
    (E) The ring of polynomials in x with real coefficients
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  2. #2
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    Quote Originally Posted by chiph588@ View Post
    (A) The ring of complex numbers
    No
    (B) The ring of integers modulo 11
    No
    (C) The ring of continuous real-valued functions on [0, 1]
    Yes. Let f(x) = \left\{ \begin{array}{c} 0 \text{ for }0\leq x \leq \tfrac{1}{2} \\ x - \frac{1}{2} \text{ for }\tfrac{1}{2} \leq x \leq 1 \end{array} \right. and g(x) = \left\{ \begin{array}{c} -x + \tfrac{1}{2} \text{ for }0\leq x \leq \tfrac{1}{2}  \\ 0 \text{ for }\tfrac{1}{2} \leq x \leq 1 \end{array} \right.
    (D) The ring {  a+b \sqrt{2} :a and b are rational numbers}
    I do not think so. You need to show if (a+b\sqrt{2})(c+d\sqrt{2}) = 0 then a=b=0 or c=d=0 by expanding out and comparing coefficients.
    (E) The ring of polynomials in x with real coefficients
    No
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