# rings

• October 30th 2008, 12:39 PM
chiph588@
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
• October 30th 2008, 01:12 PM
ThePerfectHacker
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Originally Posted by chiph588@
(A) The ring of complex numbers

No
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(B) The ring of integers modulo 11
No
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(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 $g(x) = \left\{ \begin{array}{c} -x + \tfrac{1}{2} \text{ for }0\leq x \leq \tfrac{1}{2} \\ 0 \text{ for }\tfrac{1}{2} \leq x \leq 1 \end{array} \right.$
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(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.
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(E) The ring of polynomials in x with real coefficients
No