How to show the canonical homomorphism $A_{\mathfrak p}\to B_{\mathfrak q}$ is injective?

Question

Let $A,B$ be commutative rings with identity and $f:A\to B$ an injective homomorphism of rings. For any prime ideal $\mathfrak q$ of $B$, denote $f^{-1}(\mathfrak q)$ by $\mathfrak p$, how to show the canonical homomorphism $A_{\mathfrak p}\to B_{\mathfrak q}$ is injective?

Answer 1

This is false. For instance, let $k$ be a field, let $A=k[x]$ and $B=k[x,y]/(xy)$, and let $f:A\to B$ be the obvious inclusion. Then the ideal $\mathfrak{q}=xB\subset B$ is prime, and the induced map $A_{f^{-1}(\mathfrak{q})}\to B_\mathfrak{q}$ is not injective: $x$ is nonzero in $A_{f^{-1}(\mathfrak{q})}$ but is $0$ in $B_\mathfrak{q}$ since $xy=0$ and $y\not\in\mathfrak{q}$.