Is the following module projective?

Question

Is $\Bbb Z(p^{\infty})$ a projective $\Bbb Z$-module?

Answer 1

We know that $\mathbb Z(p^{\infty})$ is an infinite $p$-primary abelian group such that each of its subgroups is finite and cyclic and has the following presentation: $$\mathbb Z(p^{\infty})=\langle x_0,x_1,x_2,...,x_n,...\mid px_0,x_0-px_1,x_1-px_2,...,x_{n-1}-px_n,...\rangle$$ so it is not free abelian with basis $X$. Because if it want to be free then it shold no relation. I think there is an exercise in An Introduction to the theory of groups that says If $G$ wants to have the projective property so it is free abelian. And this exercise shows that the Prüfer group as a $\mathbb Z-$ module can't be projective.