A ring is noetherian iff the category of finitely generated -modules is a full abelian subcategory of the category of all -modules.

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#
module theory

# Categorical noetherian-ness

# Module structures

# Freedom

A ring is noetherian iff the category of finitely generated -modules is a full abelian subcategory of the category of all -modules.

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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.