Homomorphisms from finitely generated modules and flat modules

Question

In Abstract Algebra by Dummit and Foote, question 24 in section 10.5 asks the reader to prove the equality of these statements: (1) $A$ is a flat $R$-module and (2) for any injective $R$-module homomorphism $\phi : L \rightarrow M$ with $L$ finitely generated, $1\otimes \phi : A\otimes _R L \rightarrow A\otimes _R M$ is also injective. (1) clearly implies (2) by definition but the converse isn’t as obvious. My thoughts are to show that every module can be embedded in a finitely generated module so that we can always restrict $1\otimes \phi$ to give an injective homomorphism from a module that isn’t necessarily finitely generated, yet I don’t think this is the right approach. Sorry if this is a trivial question, I just don’t want to continue without understanding this. Thanks for any help in advance.

Answer 1

Let's call a module satisfying condition 2 finitely flat. We need to prove that finitely flat modules are flat.

We may as well suppose that $L$ is a submodule of $M$. Let $A$ be finitely flat, and suppose that $\alpha$ is an element of $A\otimes L$ which becomes zero in $A\otimes M$. We can write $A=\sum_{i=1}^n r_i a_i\otimes l_i$ where $r_i\in R$, $a_i\in A$ and $l_i\in L$. So we can replace $L$ by the submodule $L'$ generated by the $l_i$.

By finite flatness, $\alpha=0$ in $A\otimes L'$ and so $\alpha=0$ in $A\otimes L$. Therefore $A$ is flat.