000 a
999 _c33981
_d33981
008 250531b xxu||||| |||| 00| 0 eng d
020 _a9783031569098
082 _a512.24
_bBAN
100 _aBandini, Andrea
245 _aCommutative Algebra through Exercises
260 _bSpringer,
_c2024
_aCham :
300 _axi, 392 p, ;
_bill.,
_c24 cm
365 _b64.99
_c
_d100.40
490 _aUNITEXT La Matematica per il 3+2, 2038-5757 ;
_vv.159
504 _aIncludes bibliographical references and index.
520 _aThis 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.
650 _aNoetherian Rings
650 _aCommutative Algebra
650 _aTensor Products
650 _aA-module
650 _aDirect sum
650 _aFree module
650 _aGröbner basis
650 _aMaximal ideal
650 _aMonomial ordering
650 _aPrime ideal
650 _aRing homomorphism
700 _aGianni, Patrizia
700 _aSbarra, Enrico
942 _2ddc
_cBK