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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 $\displaystyle f(x) = \left\{ \begin{array}{c} 0 \text{ for }0\leq x \leq \tfrac{1}{2} \\ x - \frac{1}{2} \text{ for }\tfrac{1}{2} \leq x \leq 1 \end{array} \right.$ and $\displaystyle g(x) = \left\{ \begin{array}{c} -x + \tfrac{1}{2} \text{ for }0\leq x \leq \tfrac{1}{2} \\ 0 \text{ for }\tfrac{1}{2} \leq x \leq 1 \end{array} \right.$

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(D) The ring {$\displaystyle a+b \sqrt{2} $ :a and b are rational numbers}

I do not think so. You need to show if $\displaystyle (a+b\sqrt{2})(c+d\sqrt{2}) = 0$ then $\displaystyle a=b=0$ or $\displaystyle c=d=0$ by expanding out and comparing coefficients.

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(E) The ring of polynomials in x with real coefficients

No