Flatness via factoring homomorphisms

Question

Theorem (4.32) of "Lectures on modules and rings" by T.Y. Lam says that a module $P_R$ is flat iff any $R$-homomorphism $λ:M→P$ where $M$ is any finitely presented $R$-module can be factord through a finitely generated free module: there exist $ν:M→R^m , μ:R^m→P$ (for some finite $m$) with $λ=μoν$. My question leans on the 'if" part if one wants to use Theorem (4.24)(3). I could not realize how to reach a finitely presented module $M$ and a $λ$ from the hypothesis of Theorem (4.24)(3). Thanks for any help!

Answer 1

Assume you have relations $ \sum_j a_j r_{j,l} = 0 $ with $a_j \in P$, $r_{j,l} \in R$, $1 \le j \le n$, and $1 \le l \le p$, as in Theorem 4.24(3).

Let $e_j$ denote the $j$-th standard unit vector of $R^n$. Now let $K$ be the submodule of $R^n$ generated by $\{ \sum_{j} e_j r_{j,l} \mid 1 \le l \le p\}$. Set $M=R^n/K$ and let $\lambda \colon M \to P$ be the homomorphism induced from the homomorphism $R^n \to P$, $e_j \mapsto a_j$. ($K$ is indeed in the kernel of this homomorphism due to the given relations.)

Can you take it from there?