nLab pointed module

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Definition

Definition

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

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

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

Remark

Equivalently, a pointed RR-module (Def. ) is an RR-module MM equipped with an RR-linear map RMR \to M (which is fixed by its image of 1R1 \in R). Since RR is the tensor unit for the tensor product of modules in the monoidal category R Mod R \mathrm{Mod} of RR-modules and RR-linear maps, this means that pointed RR-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 RRModR \downarrow R \mathrm{Mod} of R Mod R \mathrm{Mod} under (the underlying RR-module of) RR.

Last revised on August 7, 2023 at 14:20:16. See the history of this page for a list of all contributions to it.