A ring is noetherian iff the category of finitely generated -modules is a full abelian subcategory of the category of all -modules.
Consider the group ring . How many module structures, up to isomorphism, does admit? Do the same for . With this information, calculate in such cases.
Every morphism of abelian groups vanishing on is identically zero.
Use this fact to prove that is not a free abelian group.