Flat Non Projective $A$-Module

Question

A standard fact in Commutative Algebra is that a Projective $A$-module is flat.

The converse is false.

Can someone show me an example of a Flat Non Projective $A$-Module?

Thank you!

Answer 1

An elementary proof that $\mathbf Q$ is not projective over $\mathbf Z$: if $\mathbf Q$ were projective, it would be a direct summand of a free $\mathbf Z$-module $L$, hence there would be an injective homomorphism from $\mathbf Q$ into $L$.

However the only homorphism from $\mathbf Q$ into a free module is the null homomorphism: indeed, for any $n$, and any homomorphism $f\colon \mathbf Q\to L$, we have $$f(1)=2^nf\Bigl(\dfrac1{2^n}\Bigr)\in 2^n L,\quad\text{hence}\quad f(1)\in\bigcap_{n\ge 0}2^n L=0.$$ Since $f(1)=0$, it is easy to deduce $f\Bigl(\dfrac ab\Bigr)=0\;$ for any $\;\dfrac ab\in\mathbf Q$.

Answer 2

The rationals as a module over the integers is flat but not projective.