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020 _a9789819752843
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024 7 _a10.1007/978-981-97-5284-3
_2doi
050 4 _aQA251.3
072 7 _aPBF
_2bicssc
072 7 _aMAT002010
_2bisacsh
072 7 _aPBF
_2thema
100 1 _aWang, Fanggui.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
_983688
245 1 0 _aFoundations of Commutative Rings and Their Modules
_h[electronic resource] /
_cby Fanggui Wang, Hwankoo Kim.
250 _a2nd ed. 2024.
264 1 _aSingapore :
_bSpringer Nature Singapore :
_bImprint: Springer,
_c2024.
300 _aXXI, 847 p. 1 illus.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aAlgebra and Applications,
_x2192-2950 ;
_v31
505 0 _aBasic 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.
520 _aThis 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.
650 0 _aCommutative algebra.
_970151
650 0 _aCommutative rings.
_970152
650 0 _aAlgebra, Homological.
_970150
650 1 4 _aCommutative Rings and Algebras.
_970156
650 2 4 _aCategory Theory, Homological Algebra.
_970155
700 1 _aKim, Hwankoo.
_eauthor.
_0(orcid)0000-0002-3563-077X
_1https://orcid.org/0000-0002-3563-077X
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
_983689
710 2 _aSpringerLink (Online service)
773 0 _tSpringer Nature eBook
776 0 8 _iPrinted edition:
_z9789819752836
776 0 8 _iPrinted edition:
_z9789819752850
776 0 8 _iPrinted edition:
_z9789819752867
830 0 _aAlgebra and Applications,
_x2192-2950 ;
_v31
_983690
856 4 0 _uhttps://doi.org/10.1007/978-981-97-5284-3
912 _aZDB-2-SMA
912 _aZDB-2-SXMS
999 _c63095
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