When I say module here I mean a left module
I am trying to prove that if there is a surjective homomorphism of $R$-modules $\phi : R^n \rightarrow M $ then $M$ is finitely generated as an $R$-module.
I start by defining $x_1= \phi (1,0,...,0) , x_2 = \phi (0,1,0,...,0), ..., x_n = \phi (0,0,...,0,1) $. Then $x_i \in M $ for all $i$.
I want to show that the span of these $x_i $ is $M$.
So if $m \in M$ then $m=\phi (r_1,...,r_n )$ for some $r_1,...,r_n \in R $. I think then we should be able to say that $m= r_1 x_1+...+r_nx_n=(r_1.1,...,r_n.1 )$ but are we able to say that $r_i.1=r_i $?
If not then am I going about this the wrong way?