Definition
An R-module is called semisimple if it is isomorphic to a direct sum of simple submodules.
Equivalent conditions
The following conditions on an
is a sum (not necessarily direct) of simple submodules is semisimple - For every submodule
, there exists a submodule such that
Importance
This equivalence says that the direct part of the definition is technically not necessary (but can be convenient).
It also implies that complements of submodules, i.e. splittings of all submodules is an important structure in determining semisimplicity.
Proof
todo - Lecture 22
As a nice subcategory
Any submodule or quotient of a semisimple
is a nice subcategory.
Proof
todo - Lecture 23