Character module

The group ${\displaystyle (\mathbb {Q} /\mathbb {Z} ,+)}$, the group of rational numbers modulo ${\textstyle 1}$, can be considered as a ${\displaystyle \mathbb {Z} }$-module in the natural way. Let ${\displaystyle M}$ be an additive group which is also considered as a ${\displaystyle \mathbb {Z} }$-module. Then the group

$${\displaystyle M^{*}=\operatorname {Hom} _{\mathbb {Z} }(M,\mathbb {Q} /\mathbb {Z} )}$$

of ${\displaystyle \mathbb {Z} }$-homomorphisms from ${\displaystyle M}$ to ${\displaystyle \mathbb {Q} /\mathbb {Z} }$ is called the character group associated to ${\displaystyle M}$. The elements in this group are called characters. If ${\displaystyle M}$ is a left ${\displaystyle R}$-module over a ring ${\displaystyle R}$, then the character group ${\displaystyle M^{*}}$ is a right ${\displaystyle R}$-module and called the character module associated to ${\displaystyle M}$. The module action in the character module for ${\displaystyle f\in \operatorname {Hom} _{\mathbb {Z} }(M,\mathbb {Q} /\mathbb {Z} )}$ and ${\displaystyle r\in R}$ is defined by ${\displaystyle (fr)(m)=f(rm)}$ for all ${\displaystyle m\in M}$.[2] The character module can also be defined in the same way for right ${\displaystyle R}$-modules. In the literature also the notations ${\displaystyle M',M^{0}}$ and ${\displaystyle M^{+}}$ are used for character modules.[3][4]

Let ${\displaystyle M,N}$ be left ${\displaystyle R}$-modules and ${\displaystyle f\colon M\to N}$ an ${\displaystyle R}$-homomorphismus. Then the mapping ${\displaystyle f^{*}\colon N^{*}\to M^{*}}$ defined by ${\displaystyle f^{*}(h)=h\circ f}$ for all ${\displaystyle h\in N^{*}}$ is a right ${\displaystyle R}$-homomorphism. Character module formation is a contravariant functor from the category of left ${\displaystyle R}$-modules to the category of right ${\displaystyle R}$-modules.[3]

The abelian group ${\displaystyle \mathbb {Q} /\mathbb {Z} }$ is divisible and therefore an injective ${\displaystyle \mathbb {Z} }$-module. Furthermore it has the following important property: Let ${\displaystyle G}$ be an abelian group and ${\displaystyle g\in G}$ nonzero. Then there exists a group homomorphism ${\displaystyle f\colon G\to \mathbb {Q} /\mathbb {Z} }$ with ${\displaystyle f(g)\neq 0}$. This says that ${\displaystyle \mathbb {Q} /\mathbb {Z} }$ is a cogenerator. With these properties one can show the main theorem of the theory of character modules:[3]

Theorem (Lambek)[1]: A left module ${\displaystyle M}$ over a ring ${\displaystyle R}$ is flat if and only if the character module ${\displaystyle M^{*}}$ is an injective right ${\displaystyle R}$-module.

Let ${\displaystyle M}$ be a left module over a ring ${\displaystyle R}$ and ${\displaystyle M^{*}}$ the associated character module.

• The module ${\displaystyle M}$ is flat if and only if ${\displaystyle M^{*}}$ is injective (Lambek's Theorem[4]).[1]
• If ${\displaystyle M}$ is free, then ${\displaystyle M^{*}}$ is an injective right ${\displaystyle R}$-module and ${\displaystyle M^{*}}$ is a direct product of copies of the right ${\displaystyle R}$-modules ${\displaystyle R^{*}}$.[2]
• For every right ${\displaystyle R}$-module ${\displaystyle N}$ there is a free module ${\displaystyle M}$ such that ${\displaystyle N}$ is isomorphic to a submodule of ${\displaystyle M^{*}}$. With the previous property this module ${\displaystyle M^{*}}$ is injective, hence every right ${\displaystyle R}$-module is isomorphic to a submodule of an injective module. (Baer's Theorem)[5]
• A left ${\displaystyle R}$-module ${\displaystyle N}$ is injective if and only if there exists a free ${\displaystyle M}$ such that ${\displaystyle N}$ is isomorphic to a direct summand of ${\displaystyle M^{*}}$.[5]
• The module ${\displaystyle M}$ is injective if and only if it is a direct summand of a character module of a free module.[2]
• If ${\displaystyle N}$ is a submodule of ${\displaystyle M}$, then ${\displaystyle (M/N)^{*}}$ is isomorphic to the submodule of ${\displaystyle M^{*}}$ which consists of all elements which annihilate ${\displaystyle N}$.[2]
• Character module formation is a contravariant exact functor, i.e. it preserves exact sequences.[3]
• Let ${\displaystyle N}$ be a right ${\displaystyle R}$-module. Then the modules ${\displaystyle \operatorname {Hom} _{R}(N,M^{*})}$ and ${\displaystyle (N\otimes _{R}M)^{*}}$ are isomorphic as ${\displaystyle \mathbb {Z} }$-modules.[4]