2025-09-29

This following is a very brief summary of what happened in class on 2025-09-29.

We introduced three new constructions on modules:
1. The direct product of a pair of $R$-modules
2. The direct sum of a pair of $R$-modules
3. The sum of a pair of submodules of a given $R$-modules

We outlined the three constructions (as sets with binary operations and $R$-actions) within the framework of category theory. We talked about how the first two constructions occur in the category $R\mathbf{\text{-Mod}}$ and satisfy similar ("dual") universal properties, while the third construction happens in a different category (namely the category of submodules of a fixed module). We also mentioned the idea of extending these constructions to entire families of modules (or submodules).

Tomorrow we'll officially extend these construction to arbitrary collections of modules (or submodules), briefly outline some of the main properties of the three constructions, including when they're different and when they're the same/isomorphic.

Looking forward, before moving on to our next construction on modules (free modules) we'll introduce the notion of a natural transformation, which will finally allow us to start making many of our informal arguments and descriptors precise and formal.

Concepts

• Direct products of modules
• Direct sums of modules
• Sums of submodules
• Direct products vs. direct sums vs. sums

References

• Dummit & Foote, Abstract Algebra: Section 10.3