Algebras and modules
Model category presentations
Geometry on formal duals of algebras
Thoughout let be some ring. Write Mod for the category of module over . Write Set for the forgetful functor that sends a module to its underlying set.
For a module, a submodule of is a subset of which
is a subgroup of the underlying group (closed under the addition in );
is preserved by the -action.
Equivalently this means:
A submodule of is a module homomorphism whose underlying map of sets is an injection.
And since the injections in Mod are precisely the monomorphisms, this means that equivalently
A submodule of is a monomorphism in Mod. Hence a submodule is a subobject in Mod.
For regarded as a module over itself, a submodule is precisely an ideal of .
For a homomorphism of modules,
the kernel is a submodule of ,
the image is a submodule of .
Of free modules
Let be a ring.
A proof is in (Rotman, pages 650-651).
- Rotman Advanced Modern Algebra
Revised on September 24, 2012 23:28:02
by Urs Schreiber