2024-09-27

This following is a very brief summary of what happened in class on 2024-09-27.

We first embodied the motto "It's all about the arrows" by studying the hom-sets between given $R$ -modules, i.e., the set $Hom (M, N)$ . Although we didn't check every detail, we briefly outlined the following:

The set $Hom (M, N)$ can be given the structure of an abelian group, in which the sum of two morphisms is the morphism that sums the outputs of the two morphisms, i.e., $(f + g) (m) = f (m) + g (m)$ . The abelian structure in $N$ (and the compatibility of module morphisms with that structure) guarantee this operation makes the hom-set into an abelian group.
When the ring $R$ is commutative, we can upgrade this abelian group structure to a left $R$ -module structure by letting $R$ act on morphisms by acting on their outputs, i.e., $(r \cdot f) (m) = r \cdot f (m)$ . I forgot to mention it in class, but in order for this to actually define an $R$ -action on $Hom (M, N)$ , we need to require $R$ is commutative. (Can you find which property of being an $R$ -module requires $R$ to be commutative in this case?)
In the special case that $M = N$ , the abelian group $Hom (M, N)$ can be given the structure of a ring; here, the second "multiplicative" operation is composition of morphisms. Although not mentioned in class, when $R$ is commutative this ring can be given the structure of an $R$ -algebra, i.e., there is an $R$ -action compatible with the ring structure.

We then quickly noted that familiar constructions from groups and rings port directly over into the category of modules. Specifically, we noted that kernels and images of module morphisms exist, are defined as expected, and are submodules of the appropriate modules. You can also defined quotient modules (which, at the level of abelian groups, are exactly the same quotient groups) and there are the expected Isomorphism Theorems for modules.

Concepts

References

Dummit & Foote: Section 10.2