Ring theorey?

Question

Let R be the set of all real-valued functions defined for all real numbers under function addition and multiplication.

a. Prove R is a ring.
b. Determine all nilpotent elements of R.
c. Show that every nonzero element is a zero-divisor or a unit.

I got a. down no problem but I am having trouble with b. and c.

Answer 1

For b, Think of function such that (f(x))^n=0 for some positive integer n. Remember, this must work fo rall x values. So, the only way this can happen is when f(x) is the zero function.

For c, note that if a function is nonzero, there may still be x values such that f(x)=0. Also, (f(x))^(-1)=1/(f(x)). So, if a function does not have any x values that make it zero, then it is invertable; hence, a unit. If f does have x values that make f(x)=0, let A denote the set of x values that make f(x)=0. Then define
g(x)= 0 if x in R-A or x if x in A

Clearly, f(x)*g(x)=0.

Answer 2

b) f(x) = 0 for all x ( there is onl;y one such function , this is the zero-element )
c) there are no zero divisors !