category of left R-modules

This is the category of left modules over a ring RR. It is the prototype of an abelian category. The category of right modules is the same with the opposite ring RopR^{\mathrm{op}}, hence not listed here. The non-properties refer to the case that the ring is non-trivial, since for the trivial ring we get a trivial category which has all properties anyway. The category R−ModR{-}\mathbf{Mod} is split abelian iff RR is a semisimple ring, so usually it is not the case, which is why we have negated this property here.

Properties

Properties from the database

Deduced properties

Non-Properties

Non-Properties from the database

Deduced Non-Properties*

*This also uses the deduced properties.

Unknown properties

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Special morphisms

  • Isomorphisms: bijective RR-linear maps
  • Monomorphisms: injective RR-linear maps
  • Epimorphisms: surjective RR-linear maps