2024-10-03

This following is a very brief summary of what happened in class on 2024-10-03.

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 operations and $R$-actions), and then spent the rest of our time putting the constructions into 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 the constructions to entire families of modules (or submodules).

Tomorrow we'll briefly outline some of the main properties of the three constructions, including when they're different (the default situation) and when they're the same/isomorphic (in some special situations). We'll then return to the general idea of "universal properties" and see if we can't firm things up a bit.

Concepts

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

References

• Dummit & Foote: Section 10.3