04359nam a22005415i 4500 978-981-97-5284-3 DE-He213 20251030112937.0 cr nn 008mamaa 240903s2024 si | s |||| 0|eng d 9789819752843 978-981-97-5284-3 10.1007/978-981-97-5284-3 doi QA251.3 PBF bicssc MAT002010 bisacsh PBF thema Wang, Fanggui. author. aut http://id.loc.gov/vocabulary/relators/aut 83688 Foundations of Commutative Rings and Their Modules [electronic resource] / by Fanggui Wang, Hwankoo Kim. 2nd ed. 2024. Singapore : Springer Nature Singapore : Imprint: Springer, 2024. XXI, 847 p. 1 illus. online resource. text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda Algebra and Applications, 2192-2950 ; 31 Basic Theory of Rings and Modules -- Several Classical Module Classes in the Module Category -- Homological Methods -- Basic Theory of Noetherian Rings -- Extensions of Rings -- w-Modules over Rings -- Multiplicative Ideal Theory over Integral Domains -- Structural Theory of Milnor Squares -- Coherent Rings with Finite Weak Global Dimension -- Grothendieck Groups of Rings -- Relative Homological Algebra -- Cotorsion Theory. This book provides an introduction to the foundations and recent developments in commutative algebra. A look at the contents of the first five chapters shows that the topics covered are those usually found in any textbook on commutative algebra. However, this book differs significantly from most commutative algebra textbooks: namely in its treatment of the Dedekind–Mertens formula, the (small) finitistic dimension of a ring, Gorenstein rings, valuation overrings, the valuative dimension, and the Nagata rings. Chapter 6 goes on to present w-modules over commutative rings, as they are most commonly used in torsion theory and multiplicative ideal theory. Chapter 7 deals with multiplicative ideal theory over integral domains. Chapter 8 collects various results of pullbacks, especially Milnor squares and D + M constructions, which are probably the most important example-generating machines. In Chapter 9, coherent rings of finite weak global dimensions are probed, and the local ring of weak global dimension two is elaborated by combining homological tricks and methods of star operation theory. Chapter 10 is devoted to the Grothendieck group of a commutative ring. In particular, the Bass–Quillen problem is discussed. Finally, Chapter 11 introduces relative homological algebra, especially where the related notions of integral domains appearing in classical ideal theory are defined and studied using the class of Gorenstein projective modules. In Chapter 12, in this new edition, properties of cotorsion theories are introduced and show, for any cotorsion pair, how to construct their homology theory. Each section of the book is followed by a selection of exercises of varying difficulty. This book appeals to a wide readership, from graduate students to academic researchers interested in studying commutative algebra. Commutative algebra. 70151 Commutative rings. 70152 Algebra, Homological. 70150 Commutative Rings and Algebras. 70156 Category Theory, Homological Algebra. 70155 Kim, Hwankoo. author. (orcid)0000-0002-3563-077X https://orcid.org/0000-0002-3563-077X aut http://id.loc.gov/vocabulary/relators/aut 83689 SpringerLink (Online service) Springer Nature eBook Printed edition: 9789819752836 Printed edition: 9789819752850 Printed edition: 9789819752867 Algebra and Applications, 2192-2950 ; 31 83690 https://doi.org/10.1007/978-981-97-5284-3 ZDB-2-SMA ZDB-2-SXMS 63095 63095