Are $u_1,u_2,\cdots,u_n$ independent in $M$?

Question

Suppose $X=\operatorname{Spec}A$ and $A$ is Noetherian, $M$ is a $A$-module and the $\mathcal O_X$-module $\widetilde M$ is coherent. For some $x\in X$, if $\widetilde{M}_{x}$ is free of rank $n$ on $\mathcal O_{X,x}$, $u_1,u_2,\cdots,u_n\in M$, $M=Au_1+Au_2+\cdots+Au_n$, $(u_1)_x,(u_2)_x,\cdots,(u_n)_x$ is independent in $\widetilde{M}_{x}$ and $\widetilde{M}_{x}=\mathcal O_{X,x}(u_1)_x+\mathcal O_{X,x}(u_2)_x+\cdots+\mathcal O_{X,x}(u_n)_x$, are $u_1,u_2,\cdots,u_n$ independent in $M$?

Answer 1

No. For instance, let $A=\mathbb{Z}/(6)$, let $M=A/(2)$, and let $u_1$ be the nonzero element of $M$. Then $u_1$ is indepedendent over $A_{(3)}$ in the localization $M_{(3)}\cong M$, but $u_1$ is not independent over $A$