Is the induced map from A/fA to Ker(D)/fKer(D) injective?

Question

Let $D$ be a linear map on $k[x_1,x_2,x_3,x_4,x_5]$ to itself and $A \subseteq$ Ker$(D)$ be finitely generated as $k$-algebra. Let $f$ be a generator of $A$. Is the induced map $A/fA \rightarrow $ Ker$(D)/f$Ker$(D)$$ injective?