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＜E-Book＞
Constructive Commutative Algebra : Projective Modules Over Polynomial Rings and Dynamical Gröbner Bases / by Ihsen Yengui
(Lecture Notes in Mathematics. ISSN:16179692 ; 2138)

Edition	1st ed. 2015.
Publisher	(Cham : Springer International Publishing : Imprint: Springer)
Year	2015
Language	English
Size	VII, 271 p. 5 illus : online resource
Authors	*Yengui, Ihsen author SpringerLink (Online service)
Subjects	LCSH:Commutative algebra LCSH:Commutative rings LCSH:Mathematical logic LCSH:Computer science -- Mathematics  All Subject Search FREE:Commutative Rings and Algebras FREE:Mathematical Logic and Foundations FREE:Symbolic and Algebraic Manipulation
Notes	Projective modules over polynomial rings -- Dynamical Gr¨obner bases -- Syzygies in polynomial rings over valuation domains -- Exercises -- Detailed solutions to the exercises The main goal of this book is to find the constructive content hidden in abstract proofs of concrete theorems in Commutative Algebra, especially in well-known theorems concerning projective modules over polynomial rings (mainly the Quillen-Suslin theorem) and syzygies of multivariate polynomials with coefficients in a valuation ring. Simple and constructive proofs of some results in the theory of projective modules over polynomial rings are also given, and light is cast upon recent progress on the Hermite ring and Gröbner ring conjectures. New conjectures on unimodular completion arising from our constructive approach to the unimodular completion problem are presented. Constructive algebra can be understood as a first preprocessing step for computer algebra that leads to the discovery of general algorithms, even if they are sometimes not efficient. From a logical point of view, the dynamical evaluation gives a constructive substitute for two highly nonconstructive toolsof abstract algebra: the Law of Excluded Middle and Zorn's Lemma. For instance, these tools are required in order to construct the complete prime factorization of an ideal in a Dedekind ring, whereas the dynamical method reveals the computational content of this construction. These lecture notes follow this dynamical philosophy HTTP:URL=https://doi.org/10.1007/978-3-319-19494-3

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