for which of the following rings is it possible for the product of two non-zero elements to be zero?

a. ring of complex numbers

b. ring of integers modulo 11

c. ring of continuous real-valued functions on [0,1]

d. ring {a+b(2^1/2): a, b are rational numbers}

e. ring of polynomials in x with real coefficients

answer is c.

could someone explain please?

Thanks a lot.

## GR0568 Probelm 40

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.