Can you suggest some challenging ring theory problems?

Question

I would like to work on some wonderful excercises (not particularly hard) in ring theory, subjects PID, UFD or polynomial rings. Can you introduce a resource that has newer exercises? Or if you can remember a good one, please let us know about it too! :)

Thank you very much!

Answer 1

There are lots of places to find exercises. Any general algebra textbook worth its salt will have plenty of exercises on ring theory (in particular the subjects you've named).

Off my bookshelf, there's (in approximately increasing order of difficulty):

• Dummit and Foote's "Abstract Algebra" (chs 8 and 9)
• Aluffi's "Algebra: Chapter 0" (ch V.2)
• Lang's "Algebra" (ch II.5)

Again, though, almost any algebra book will do. All three of these are available (perhaps unscrupulously) in PDF form online, and (for better or for worse) complete or partial solutions to many problems (certainly from Dummit and Foote) are available on this website. You might also have luck looking at Number Theory textbooks (though I don't know much number theory, so I don't have any recommendations).

---

I hope this helps ^_^

Answer 2

Perhaps this is on the easier side (not sure what difficulty you're looking for), but sections 34 through 41 of Durbin's "Modern Algebra: An Introduction" are dedicated almost entirely to the properties of polynomial rings and polynomial quotient rings with the intent of making a beeline to some intro Galois theory. It starts fairly slowly and approachable. It's full of all kinds of exercises on polynomial rings and theorems important to the subject. There seem to be solutions online to many problems, too.

It's my favorite algebra reference, and always one of my first go-to's. Can't recommend it enough.