# Categorical noetherian-ness

A ring $A$ is noetherian iff the category $_A\mathrm{mod}$ of finitely generated $A$-modules is a full abelian subcategory of the category $_A\mathrm{Mod}$ of all $A$-modules.

Consider the group ring $\mathbb Z C_p$. How many module structures, up to isomorphism, does $C_{p^2}$ admit? Do the same for $C_p^2$. With this information, calculate $H^2(C_p,-)$ in such cases.
Every morphism $f:\Bbb{Z}^\Bbb{N}\to A$ of abelian groups vanishing on $\Bbb{Z}^{(\Bbb{N})}$ is identically zero.
Use this fact to prove that $\Bbb{Z}^\Bbb{N}$ is not a free abelian group.