Let R be the set of all real valued functions defined for all reals under function addition and multiplication. Determine all zero divisors of R. A zero divisor is a non zero element such that when multiplied with another nonzero element the product is zero. So I said that the zero divisors of R would be all the functions in R that are not the zero function but take on the value of zero at least one time. Am I close? acoording to my answer though f=sin X and g= x-2 would be zero divisors because niether function is the zero function but fg = (sin X)(x-2) = 0 at x=2 but my teacher said this is wrong he said the product has to be zero for all x. How would I even begin to find all the functions that are not the zero function but when multiplied together is zero for all x?