For (a) choose two function f,g different from the zero function but its
product fg is zero. this must be pretty easy and in fact you can choose
f,g to be differentiable
For (b): I thought F(R)* denotes the units in F(R) but then I saw UF(R) and I am not sure anymore. Anyway, the units in F(R) are those functions which have a multiplicative inverse, so f is a unit <==> we can form 1/f.
Now, when can we form 1/f? Can you see it?