04436nam a22005055i 4500 978-3-031-56910-4 DE-He213 20251030112624.0 cr nn 008mamaa 240712s2024 sz | s |||| 0|eng d 9783031569104 978-3-031-56910-4 10.1007/978-3-031-56910-4 doi QA150-272 PBF bicssc MAT002000 bisacsh PBF thema Bandini, Andrea. author. aut http://id.loc.gov/vocabulary/relators/aut 80389 Commutative Algebra through Exercises [electronic resource] / by Andrea Bandini, Patrizia Gianni, Enrico Sbarra. 1st ed. 2024. Cham : Springer International Publishing : Imprint: Springer, 2024. XI, 392 p. online resource. text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda La Matematica per il 3+2, 2038-5757 ; 159 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. This book provides a first introduction to the fundamental concepts of commutative algebra. What sets it apart from other textbooks is the extensive collection of 400 solved exercises, providing readers with the opportunity to apply theoretical knowledge to practical problem solving, fostering a deeper and more thorough understanding of the subject. The topics presented here are not commonly found in a single text. Consequently, the first part presents definitions, properties, and results crucial for understanding and solving the exercises, serving also as a valuable reference. The second part contains the exercises and a section with "True or False?" questions, which serves as a valid self-assessment test. Considerable effort has been invested in crafting solutions that provide the essential details, aiming for a well-balanced presentation. We intend to guide students systematically through the challenging process of writing mathematical proofs with formal correctness and clarity. Our approach is constructive, aiming to illustrate concepts by applying them to the analysis of multivariate polynomial rings and modules over a principal ideal domain (PID) whenever feasible. Algorithms for computing these objects facilitate the generation of diverse examples. In particular, the structure of finitely generated modules over a PID is analyzed using the Smith canonical form of matrices. Furthermore, various properties of polynomial rings are investigated through the application of Buchberger’s Algorithm for computing Gröbner bases. This book is intended for advanced undergraduates or master’s students, assuming only basic knowledge of finite fields, Abelian groups, and linear algebra. This approach aims to inspire the curiosity of readers and encourages them to find their own proofs while providing detailed solutions to support their learning. It also provides students with the necessary tools to pursue more advanced studies in commutative algebra and related subjects. Algebra. 1501 Algebra. 1501 Gianni, Patrizia. author. aut http://id.loc.gov/vocabulary/relators/aut 80390 Sbarra, Enrico. author. aut http://id.loc.gov/vocabulary/relators/aut 80391 SpringerLink (Online service) Springer Nature eBook Printed edition: 9783031569098 Printed edition: 9783031569111 La Matematica per il 3+2, 2038-5757 ; 159 70145 https://doi.org/10.1007/978-3-031-56910-4 ZDB-2-SMA ZDB-2-SXMS 61517 61517