Projective but not free module over groupring

Question

Let $G$ be a nontrivial finite group and consider the groupring $\Bbb QG$. My question is whether we can find a module over $\Bbb QG$ that is projective but not free?

Answer 1

By Maschke's Theorem, $\mathbb{Q}G$ is semisimple. In particular, every $\Bbb{Q}G$-module is projective. Thus, what you are looking for is any non-free module.

Answer 2

Any nontrivial direct summand $T$ of the right module $\Bbb Q G$, if such $T$ exists, is projective ( since summands of free modules are projective) but not free, since it can't have the dimension of a free module (which is clearly $|G|m$ where $m$ is a nonnegative integer. This argument applies to any finite group (other than $\{1\}$) and any field.

For fields of characteristic $0$ (like $\Bbb Q$) Maschke's theorem allows us to use any nontrivial right ideal at all, since they are all summands.