Is there an easy example of a module which is not projective

Question

Is there an easy example of a module which is not projective?

I found that direct product of $\mathbb Z$ is not projective but its proof is complicated.

Are there any easy examples for this?

Answer 1

Since $\Bbb Z$ is a PID and projective modules over PIDs are free, it is clear that every finite abelian group is not projective.

But in fact, you can just use the definition to prove that $\Bbb Z/2\Bbb Z$ is not projective as $\Bbb Z$-module: consider the quotient map $$ \pi:\Bbb Z/4\Bbb Z\longrightarrow\Bbb Z/2\Bbb Z. $$ It is easy to check that there is no map $f:\Bbb Z/2\Bbb Z\rightarrow\Bbb Z/4\Bbb Z$ such that $\pi\circ f=1_{\Bbb Z/2\Bbb Z}$.

Answer 2

$\mathbb Z/2\mathbb Z{}{}{}{}{}{}{}{}$.

Answer 3

Any projective module over a PID (e.g., $\mathbb{Z}$) must be free.