category of left modules over a ring

 algebra
Notation R−ModR{-}\Mod Objects left RR-modules Morphisms RR-linear maps Children R−ModR{-}\Mod, R−ModR{-}\Mod Related M−SetM{-}\Set, Ab\Ab, VectK\Vect_K, grModG(R)\grMod_G(R), Projfg(R[ε])\Proj_\fg(\IR[\varepsilon]), Freefg(Z×Z)\Free_{\fg}(\IZ \times \IZ) External nLab Link

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. We assume R≠0R \neq 0 since otherwise the category would be trivial.

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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Undecidable properties

There is 1 property for which it cannot be decided if it is satisfied or not.

Special objects

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

Special morphisms

  • isomorphisms: bijective morphisms
  • monomorphisms: injective morphisms
  • epimorphisms: surjective morphisms
  • regular monomorphisms: same as monomorphisms
  • regular epimorphisms: surjective morphisms

Symmetric monoidal categories

The database stores 3 symmetric monoidal categories based on the category of left modules over a ring.