There is a question and an answer on the following link: Question concerning a faithful module over an Artinian ring
But there is one place I can't understand in the answer of the question. That is :why this map $A \rightarrow \Pi _{i \in I} (Ax_i)$ given by $a \rightarrow (ax_i)_{i\in I}$ is injective? Can anyone tell me? Thank you.
That is a part of the proof of the $\implies$ direction, and therefore it is being assumed that $M$ is a faithful module, i.e. that $\mathrm{Ann}_A(M)=0$. Also, note that $(x_i)_{i\in I}$ generates $M$.
Assume that for some $a\in A$, we have $ax_i=0$ for all $i\in I$. Then $am=0$ for all $m\in M$ since the $x_i$'s are a generating set, and thus $a\in\mathrm{Ann}_R(M)$, hennce $a=0$. Therefore the map is injective.