category of finitely generated free abelian groups

 algebra
Notation FreeAbfg\FreeAb_\fg Objects finitely generated free abelian groups Morphisms group homomorphisms Related Ab\Ab, TorsFreeAb\TorsFreeAb, Abfg\Ab_{\fg}, FreeAb\FreeAb, FinVectK\FinVect_K, Projfg(R[ε])\Proj_\fg(\IR[\varepsilon]), Freefg(Z×Z)\Free_{\fg}(\IZ \times \IZ)
This is the full subcategory of Ab\Ab consisting of the free abelian groups that are also finitely generated. Equivalently, it is the category Projfg(Z)\Proj_\fg(\IZ) of finitely generated projective Z\IZ-modules. Every object is isomorphic to Zn\IZ^n for a unique n∈Nn \in \IN.

Satisfied properties

Assigned properties

Deduced properties

Unsatisfied properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

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Special objects

  • terminal object: trivial group
  • initial object: trivial group
  • products: [finite case] direct sums
  • coproducts: [finite case] direct sums

Special morphisms

  • isomorphisms: bijective homomorphisms
  • monomorphisms: injective homomorphisms
  • epimorphisms: homomorphisms f:A→Bf : A \to B such that f(A)f(A) is not contained in a proper direct summand of BB
  • regular monomorphisms: injective homomorphisms f:A→Bf : A \to B such that B/f(A)B/f(A) is torsion-free (and these are automatically split monomorphisms)
  • regular epimorphisms: surjective homomorphisms (which are automatically split epimorphisms)