Let R be a commutative ring with identity $\displaystyle 1 ( \neq 0 ) $ and let $\displaystyle f(x)= f_nx^n+ . . . + f_1x+f_0$ be a polynomial in $\displaystyle R[x]$

Prove that $\displaystyle f(x)$ is a zero divisor of $\displaystyle R[x]$ if and only if there is a nonzero element $\displaystyle s \in R $ such that $\displaystyle sf(x) = (0) $

Proof so far.

Suppose that $\displaystyle f(x)$ is a zero divisor, then by definition $\displaystyle \exists g(x) \in R[x], g(x) \neq 0 $ such that $\displaystyle f(x)g(x) = 0$

But how would I factor this down into one single element?

Conversely, suppose that $\displaystyle sf(x)=(0) \ \ \ s \in R $, then would I be able to find a polynomial $\displaystyle g(x)$ that behaves like s?

Thank you!