REU Meeting - 2025-07-10

This following is a brief summary of our research meeting on 2025-07-10.

Meeting summary

We spent our time going through solutions to exercises 13 and 14 in Section 18.1 of Dummit & Foote. We left hanging one important detail, namely proving the following parenthetical claim made in part (c) of Problem 14:

We also started working on Exercises 15 and 16, but we didn't fully flesh out determining which representations are inequivalent.

Tasks for next meeting

• Restate the above exercise more formally and precisely, and then find a proof.
• Rephrase that result more categorically. After all, what does it even mean to say "an abelian subgroup of a division ring"? Those structures are in different categories.
• In exercises 15 and 16, explicitly translate what it means for two of your representations to be equivalent, and then determine which are equivalent and which are not.
• Review the notions of "direct sum" and "submodule" in the following two categories: 1) the category of all left $R$-modules; and 2) the category of submodules of a given $R$-module, $M$. Give categorical descriptions of each.

References

Dummit & Foote: Section 18.1