In mathematics, especially in the area of abstract algebra, every module has an associated character module. Using the associated character module it is possible to investigate the properties of the original module. One of the main results discovered by Joachim Lambek shows that a module is flatif and only if the associated character module is injective.
Definition
The group, the group of rational numbers modulo, can be considered as a -module in the natural way. Let be an additive group which is also considered as a -module. Then the group of -homomorphisms from to is called the character group associated to . The elements in this group are called characters. If is a left -module over a ring, then the character group is a right -module and called the character module associated to. The module action in the character module for and is defined by for all. The character module can also be defined in the same way for right -modules. In the literature also the notations and are used for character modules. Let be left -modules and an -homomorphismus. Then the mapping defined by for all is a right -homomorphism. Character module formation is a contravariant functor from the category of left -modules to the category of right -modules.
Motivation
The abelian group is divisible and therefore an injective -module. Furthermore it has the following important property: Let be an abelian group and nonzero. Then there exists a group homomorphism with. This says that is a cogenerator. With these properties one can show the main theorem of the theory of character modules:
Theorem : A left module over a ring is flat if and only if the character module is an injective right -module.
Properties
Let be a left module over a ring and the associated character module.
For every right -module there is a free module such that is isomorphic to a submodule of. With the previous property this module is injective, hence every right -module is isomorphic to a submodule of an injective module.
A left -module is injective if and only if there exists a free such that is isomorphic to a direct summand of.
The module is injective if and only if it is a direct summand of a character module of a free module.
If is a submodule of, then is isomorphic to the submodule of which consists of all elements which annihilate.
