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    Ring proof

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    Let R = C([0, 1]) be the ring of continuous real-valued functions on the interval [0, 1], with the usual definitions of sum and product of functions from calculus.
    Show that f in R is a zero divisor if and only if f is not identically zero and { x | f(x) = 0 } contains an open interval. What are the idempotents of this ring? What are the nilpotents? What are the units?

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    Solution Summary

    This solution contains a proof regarding a ring and divisors.