This following is a very brief summary of what happened in class on 2025-09-25.
We spent a good portion of the first half of class analyzing the hom-sets , i.e., the set of morphisms between two fixed -modules. We noted (and mostly verified) that this set has a (natural) structure of an abelian group, and that when is commutative, even the structure of an -module itself. And in the special case where , the set of endomorphisms of has the structure of a ring (and when is commutative, an -algebra). These observations hint at an idea known as "enriched category theory," which we sadly won't have time to explore.
We then moved on to talk about quotient modules. Fortunately they match exactly with the usual construction of quotient groups in , just with an added -action thrown in. We ended by talking about a "universal property" of the quotient module construction, an idea we'll explore much more deeply soon.