The polynomial ring of a right duo ring is semicommutative?

Question

A ring $R$ is said to be right duo if every right ideal is two sided. A ring $R$ is said to semicommutative if $a,b\in R$ satisfy $ab=0$ then $aRb=(0)$. I am thinking about whether the polynomial ring of a right duo ring is semicommutative.
Proof Attempt: Let $R$ be a right duo ring. Take $f(x),g(x)\in R[x]$ such that $f(x)g(x)=0$. We have to prove that $f(x)h(x)g(x)=0$ $\forall h(x) \in R[x]$. How to go further, should I expand $f(x),g(x)$?