I am having difficulty to prove that commutative rings are McCoy. It means if $R$ is any commutative ring and $R[x]$ is its polynomial ring, whenever two polynomials $f(x),g(x)$ $\in R[x]$ annihilate each other means $f(x)g(x)=0$ then $\exists$ an annihilator in base ring means $\exists$ $(0 \ne r)\in R$ such that $rf(x)=0$ and $g(x)r=0$.
My approach: Suppose $R$ is any commutative ring, and take $f(x),g(x)$ $\in R[x]$ such that $f(x)g(x)=0$, so if $f(x)=\sum_{i=0}^{n}a_ix^i $ and $g(x)=\sum_{j=0}^{m}b_jx^j $, then we have the following equations
$a_0b_0=0$, $a_1b_0+a_0b_1=0$, $...$
Now, how should I go further, any idea ?
