We know that for a field $k $ and $f\in k [x]$, the only maximal ideals of $k [x]$ containing $f $ are the ideals generated by prime factors of $f $. Now, I want to know that if $R $ is an arbitrary commutative ring with identity, is there any description for maximal ideals containing $f $?

We know that for a field $k $ and $f\in k [x]$, the only maximal ideals of $k [x]$ containing $f $ are the ideals generated by prime factors of $f $. Now, I want to know that if $R $ is an arbitrary commutative ring with identity, is there any description for maximal ideals containing $f $?Or, can we find $\cap_{f\in m\in Max (R [x])}m$?