Intersection of ideal in (non-noetherian) polynomial ring

Question

Let $A$ be a (commutative, associative, unital) ring. Let $f$ be a non-zerodivisor in $A$, and let $I$ be a finitely-generated ideal in $A[\frac1f][T_1,\dotsc,T_r]$. Can it happen that $I\cap A[T_1,\dotsc,T_r]$ is no longer a finitely-generated ideal in $A[T_1,\dotsc,T_r]$? By the Hilbert basis theorem, this cannot happen when $A$ is noetherian, but I'm curious about the general case. Thanks!