$R$ is left noetherian. $A$ is finitely generated $R$-module. Then $A$ is finitely generated free.
I can't see why it is free. I know that $A$ is finitely presented, so there is an exact sequence:
$0 \to P \to F\to A \to 0$ where $F$ is free and both $F, P $ are finitely generated. But a quotient of free module is not necessarily free, right? So how is $A$ free? Any help would be apprecited!