category of left modules over a ring

This is the prototype of an abelian category. The category of right modules is the same with the opposite ring RopR^{\op}, hence not listed here.
To settle the unsatisfied properties, we make the assumption that RR is not semisimple: If RR is semisimple, then by the Artin-Wedderburn theorem, the category is equivalent to a finite direct product of categories D−ModD{-}\Mod for division rings DD, and the case of division rings is in a separate entry. In particular, R≠0R \neq 0 and RR is not a field.

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: zero module
  • initial object: trivial module
  • products: direct products with pointwise operations
  • coproducts: direct sums

Special morphisms

  • isomorphisms: bijective RR-linear maps
  • monomorphisms: injective RR-linear maps
  • epimorphisms: surjective RR-linear maps
  • regular monomorphisms: same as monomorphisms
  • regular epimorphisms: surjective homomorphisms

Undistinguishable categories

These categories in the database currently have exactly the same properties as the category of left modules over a ring. This indicates that the data may be incomplete or that a distinguishing property may be missing from the database.