Finitely generated modules and their quotients

Question

I am a bit puzzled by a line in a proof. It says that a finitely generated $R$-module $M$ is 'a quotient module of some $R^k$ '.

Does that mean that $M = R^k/L$ (or just isomorphic) for some submodule $L$ of $R^k$? I cannot see why this is the case and would appreciate some pointers/an explanation.

Thanks

Answer 1

Suppose $M$ is generated by $m_1, \dots,m_k$. Let $(e_1)_{1\le i\le k}$ be the canonical basis of $R^k$ and consider the linear map \begin{align} p:R^k&\longrightarrow M,\\ e_i&\longmapsto m_i. \end{align} This is clearly surjective, hence if $L=\ker p$, we have $R^k/L\simeq M$ by the first isomorphism theorem.

Answer 2

Isomorphic. There is a surjection $R^k\to M$.