Do we have $\ker (f)_{\mathfrak p}\simeq \ker (g)$?

Question

Let $A,B$ be commutative rings with identity and $f:A\to B$ a homomorphism of rings. For any prime ideal $\mathfrak q$ of $B$, denote $f^{-1}(\mathfrak q)$ by $\mathfrak p$, then $f$ induces the canonical homomorphism $g: A_{\mathfrak p}\to B_{\mathfrak q}$, do we have $(\ker f)_{\mathfrak p}\simeq \ker g$?