On a McCoy-like condition for rings

We study rings $R$ for which whenever non-zero polynomials $f(x)$ and $g(x)$ satisfy $f(x)g(x)f(x)=0$, it implies that there is a non-zero element $r\in R$ such that $f(x)rf(x)=0$. We call such rings inner McCoy rings. We explore some examples of rings that are inner McCoy, determine relationships between the class of inner McCoy rings and some known classes of rings. Furthermore, we show that the maximal inner McCoy subring of a matrix ring is the triangular matrix ring. Finally, we construct new inner McCoy rings from known ones.