polynomial multiplication with integer polynomial result

Question

A(z), B(s), D(s) are polynomials. A(s) is a polynomial on top of the integers, and D(s) is a monic polynomial not necessarily with integer coefficients. B(s) is not necessarily monic. I have:

A(s) = B(s)D(s)

Does that imply B(s) is on top of the integers? if so, how do I prove it?