Character module

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 flat if and only if the associated character module is injective. 
Definition.
The group formula_1, the group of rational numbers modulo formula_2, can be considered as a formula_3-module in the natural way. Let formula_4 be an additive group which is also considered as a formula_3-module. Then the group formula_6 of formula_3-homomorphisms from formula_4 to formula_9 is called the "character group associated to formula_4". The elements in this group are called "characters". If formula_4 is a left formula_12-module over a ring formula_12, then the character group formula_14 is a right formula_12-module and called the "character module associated to" formula_4. The module action in the character module for formula_17 and formula_18 is defined by formula_19 for all formula_20. The character module can also be defined in the same way for right formula_12-modules. In the literature also the notations formula_22 and formula_23 are used for character modules. 
Let formula_24 be left formula_12-modules and formula_26 an formula_12-homomorphismus. Then the mapping formula_28 defined by formula_29 for all formula_30 is a right formula_12-homomorphism. Character module formation is a contravariant functor from the category of left formula_12-modules to the category of right formula_12-modules. 
Motivation.
The abelian group formula_34 is divisible and therefore an injective formula_35-module. Furthermore it has the following important property: Let formula_36 be an abelian group and formula_37 nonzero. Then there exists a group homomorphism formula_38 with formula_39. This says that formula_34 is a cogenerator. With these properties one can show the main theorem of the theory of character modules: Theorem (Lambek) : A left module formula_4 over a ring formula_12 is flat if and only if the character module formula_14 is an injective right formula_12-module. 
Properties.
Let formula_4 be a left module over a ring formula_12 and formula_14 the associated character module. 