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For ​any textbook considered ‘standard’ in a particular discipline, there are a plethora of solutions available online to the standard exercises in these texts. I have been told many times, and I agree, that doing as many exercises as possible in introductory texts is the best way to dive into new theory. Below are select solutions to exercises in various introductory commutative algebra texts.

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Important Note: These exercises were used to aid in self-study and I have posted them here to aid anyone else doing the same. I strongly caution against the misuse of them for any other purpose. Also note that I often find terse proofs more frustrating than elegant when learning a new subject, and this is reflected in the proof writing style. Lastly, these exercises and notes are handwritten, with the intention that I will TeX them up at a later date.

Commutative Ring Theory
Irving Kaplansky

Chapter 1
  • 1-1 Prime Ideals
  • 1-2 Integral Elements, I
  • 1-3 G-ideals, Hilbert Rings, and the Nullstellensatz
  • 1-4 Localization
  • 1-5 Prime Ideals in Polynomial Rings
  • 1-6 Integral Elements, II

Chapter 2
  • 2-1 The Ascending Chain Condition
  • 2-2 Zero-divisors
  • 2-3 Integral Elements
  • 2-4 Intersections of Quasi-local Domains

Chapter 3
  • 3-1 R-sequences and Macaulay Rings
  • 3-2 The Principal Ideal Theorem
  • 3-3 Regular Rings

Note: My academic brother Kevin Bombardier also has a very nice set of solutions to exercises in this text that you can find here.

Introduction to Commutative Algebra
M.F. Atiyah and I.G. MacDonald

  • Chapter 1 Rings and Ideals