Dimensions of Ring Theory
Paperback Engels 2011 9789401082075Samenvatting
Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Gad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of s9phistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.
Specificaties
Lezersrecensies
Inhoudsopgave
(M) and CHg.- 7.3. Relative Krull Dimension.- 7.4. Relative Krull Dimension Applied to the Principal Ideal Theorem.- 7.5. Relative Gabriel Dimension.- 7.6. Relative Krull and Gabriel Dimensions of Graded Rings.- 7.7. Exercises.- Bibliographical Comments to Chapter 7.- 8. Homological Dimensions.- 8.1. The Projective Dimension of a Module.- 8.2. Homological Dimension of Polynomial Rings and Rings of Formal Power Series.- 8.3. Injective Dimension of a Module.- 8.4. The Flat Dimension of a Module.- 8.5. The Artin-Rees Property and Homological Dimensions.- 8.6. Regular Local Rings.- 8.7. Exercises.- Bibliographical Comments to Chapter 8.- 9. Rings of Finite Global Dimension.- 9.1. The Zariski Topology.- 9.2. The Local Study of Homological Dimension.- 9.3. Rings Integral over their Centres.- 9.4. Commutative Rings of Finite Global Dimension.- 9.5. Exercises.- Bibliographical Comments to Chapter 9.- 10. The Gelfand-Kirillov Dimension.- 10.1. Definitions and Basic Properties.- 10.2. GK-dimension of Filtered and Graded Algebras.- 10.3. Applications to Special Classes of Rings.- 10.3.1. Rings of Differential Operators and Weyl Algebras.- 10.3.2. Remarks on Enveloping Algebras of Lie Algebras.(Addendum)..- 10.3.3. P.I.Algebras.(Addendum).- 10.4. Exercises.- Bibliographical Comments to Chapter 10.- References.
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