there are some facts you need to know :
if F is a field then there is no zero divisor in F so a, b, d are incorrect
if R is a domain then R[x] - the ring of polynomial - is a domain also so R[x] has no zero divisor - hence e is incorrect.
so the answer is c.
By the way, we can find two non-zero functions f, g in C[0,1] st : f.g=0.