TITLE: The relationship of free, divisible, projective, injective, and flat modules.
QUESTION [9 upvotes]: In general, we have that:
free $\Rightarrow$ projective $\Rightarrow$ flat
injective $\Rightarrow$  divisible ( ($\Rightarrow$) be ($\Leftrightarrow$) in PIDs)
Simple Counter-examples:
projective but not free: $\mathbb{Z}_2$ is $\mathbb{Z}_6$  projective but not $\mathbb{Z}_6$ free
flat but not projective: $\mathbb{Q}$
My questions:

1) Please give counter-examples: divisible but not injective, flat but not injective.  
2) In proof about $\mathbb{Q}$ is not a projective $\mathbb{Z}$-module, I use 2 properties:

First: $P$ is projective $\Leftrightarrow$  there is a free module $F$ and an $R$-module $K$ such that $F≅K⊕P$.
Second: Every submodule of a free module in PID is free.
The first is easy to prove but the second isn't. Other way to prove $\mathbb{Q}$ is not projective that use projective basis, but really it's difficult to understand for me.  

So is there other way?

Thanks for regarding!

REPLY [6 votes]: 1st question answered in comments. For 2nd:  

Lemma: Let $R$ be a domain s.t there is an $R$-module, $M$, which is nonzero, projective and injective $R$-module. then $R$ is a field.   

$\mathbb{Q}$ is a nonzero and injective $\mathbb{Z}$-module (as you mentioned injective $\Leftrightarrow$  divisible in  PIDs). $\mathbb{Z}$ is a domain and not field. now use lemma.