Commutative Algebra through Exercises

Andrea Bandini is an associate professor of algebra at the Department of Mathematics of the University of Pisa. He has taught several courses in basic algebra, commutative algebra, and number theory. His research interests mainly concern algebraic number theory and arithmetic geometry. Patrizia Gianni is a professor of algebra specializing in computer algebra and is recognized for her contributions to Gröbner bases and computational real algebraic geometry. She played a key role in the development of the Axiom computer algebra system.Enrico Sbarra received his doctorate in Germany at the University Duisburg-Essen under the supervision of Jürgen Herzog. After collaborating with the Universities of Trieste, Bochum and Genoa, since 2008, he has been a researcher in algebra and a lecturer at the Department of Mathematics in Pisa. He is the author of several papers published in prominent international journals. His research interests include combinatorial and commutative algebra, with applications to algebraic geometry.

"All proofs and solutions are complete with an occasional alternative solution offered. ... This is a good textbook for the purposes intended.” (Radoslav M. Dimitrić, zbMATH 1554.13001, 2025)

• Part I Theory.- 1 Rings.- 2 The Ring K[x1, . . . , xn].- 3 Affine Algebraic Varieties.- 4 Modules.- 5 Tensor Product.- 6 Localization.- 7 Noetherian and Artinian Rings. Primary Decomposition.- Part II Exercises.- 8 Rings and Ideals.- 9 Polynomials, Gröbner Bases, Resultant, and Varieties.- 10 Modules.- 11 Tensor Product.- 12 Localization.- 13 Noetherian and Artinian Modules.- 14 True or False?.- 15 Review Exercises.- Part III Proofs and Solutions.- 16 Proofs of Theoretical Results.- 17 Solutions to the Exercises.