nLab pointed module

Definition

Definition

Let $R$ be a (commutative) ring. Then a pointed $R$-module is an $R$-module $M$ equipped with the further choice of an element $m \in M$. A homomorphism of pointed modules is a homomorphism of the underlying modules which preserves the choice of the extra points.

If $R$ is a field, then pointed $R$-modules are also called pointed $R$-vector spaces.

If $R$ is the integers then pointed $R$-modules are also called pointed abelian groups.

Remark

Equivalently, a pointed $R$-module (Def. ) is an $R$-module $M$ equipped with an $R$-linear map $R \to M$ (which is fixed by its image of $1 \in R$). Since $R$ is the tensor unit for the tensor product of modules in the monoidal category $R \mathrm{Mod}$ of $R$-modules and $R$-linear maps, this means that pointed $R$-modules are equivalently pointed objects in the monoidal-categorical sense (in addition to their underlying sets being pointed objects in the set-theoretic sense). Accordingly, the category of pointed modules is equivalently the coslice category $R \downarrow R \mathrm{Mod}$ of $R \mathrm{Mod}$ under (the underlying $R$-module of) $R$.

Related concepts

• module

• pointed abelian group

• pointed object in a monoidal category

• functorial aspects of the GNS representation