Let $X$ be a topological space and $A$ a sheaf of rings over $X$. Let $F$ be a sheaf of $A$-modules over $X$.
The original motivation for my question is that I want to prove that locally free sheaves are torsion-free. For that matter I am arguing by contradiction, assuming that we have some section $s\in F(U)$ and some element $a\in A(U)$ such that $a \cdot s = 0$ in $F(U)$. I want to use that $F$ is locally free and partition $U$ with $U\cap U_\alpha$ where $\{U_\alpha\}$ is an open covering of $X$ such that $$ F(U_\alpha) \simeq A(U_\alpha)^r $$ for some constant $r$ (assuming $X$ is connected). However, I cannot close the following diagram from below $\require{AMScd}$
\begin{CD} F(U_\alpha) @>{\sim}>> A(U_\alpha)^r\\ @V{\rho}VV @V{\rho}VV\\ F(U\cap U_\alpha) @. A(U\cap U_\alpha) \end{CD} which would allow me to easily reach a contradiction.
Any ideas on how to prove this?