I am studing the book "Rings and Categories of modules" written by Frank W. Anderson and Kent R. Fuller. On page 124, I am at a loss for an example.
Example. The abelian group $\mathbb{Z}$ is finitely generated but not finitely cogenerated. The abelian group $\mathbb{Z}_{p^\infty}$ is finitely cogenerated but not finitely generated.
I post my effort here.
(1).We can regard them as modules over $\mathbb{Z}$.
(2).$\mathbb{Z}$ is finitely generated is clear.
(3). By the chain $0 \subset \left({1 \over p}\mathbf{Z}\right)/\mathbf{Z} \subset \left({1 \over p^2}\mathbf{Z}\right)/\mathbf{Z} \subset \left({1 \over p^3}\mathbf{Z}\right)/\mathbf{Z} \subset \cdots$ of $\mathbb{Z}_{p^\infty}$. We know $\mathbb{Z}_{p^\infty}$ as $\mathbb{Z}$-module is Artinian but not Noetherian.
Any help will be appreciated.
Ad (1): You mean you regard them both as modules over ${\mathbb Z}$, no?
Ad (3): The chain you provide indeed shows that ${\mathbb Z}_{p^{\infty}}$ is not finitely generated, but to show that it is artinian and hence finitely cogenerated, you still need to note that you've actually listed all submodules.