Exercises for commutative algebra

Question

I'm currently studying for my final exam of commutative algebra. In the class we covered Artinian rings, Dedekind domains, Integral closures, Grobner basis, (discrete )Valuation rings... I'm particularly interested in exercises involving the exactness of $\operatorname{Hom}(\cdot,N) $ and $\operatorname{Hom}(N,\cdot) $.

Thought that you guys could provide some exercises.

Thank you all in advance.

Answer 1

A few suggestions:

1. The canonical recommendation is probably Commutative Algebra by Atiyah and MacDonald. There are a lot of exercises and you can find solutions online to most of them if you want to check your work.
2. Lang's Algebra has many sections that might be useful. In particular, his second chapter contains a section of exercises on Dedekind Domains, as does the third chapter if I recall correctly.
3. If you are looking for some somewhat easier exercises, maybe you can try some of the material in Dummit and Foote.
4. I believe that Eisenbud's book Commutative Algebra with a View Towards Algebraic Geometry goes into some depth on Gröbner Bases.

My own personal experience is mostly with the first three suggestions. I did a large portion of the exercises in the first 5 chapters of Atiyah and MacDonald and a fair few exercises in Lang.