2025-09-26

This following is a very brief summary of what happened in class on 2025-09-26.

We began by recapping a universal property of the quotient module construction. I made the claim that this property uniquely characterizes the morphism π:MM/N and that all properties of M/N can be deduced from it. As an illustration, we saw how one can prove the Third Isomorphism Theorem for Modules with "for free" with repeated use of our universal property. (Once we have the corresponding universal properties for kernels and images we can give an even cleaner proof, without any references whatsoever to elements!)

We then explored the idea of a "universal property" for other constructions, looking at several other examples, some familiar and others less so. We will return to this idea several times over the next few weeks, gradually filling in the details (including eventually a precise, formal definition for "universal property").

Concepts


Quotient modules
The Isomorphism Theorems for Modules
Universal Properties I - Inspiring Examples

References