Assignment:
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?
Provide complete and step by step solution for the question and show calculations and use formulas.