06948nmm a2200409 u 4500001001200000003002700012005001700039007002400056008004100080020001800121100001900139245009500158250001700253260004800270300003100318505096100349505022201310505094001532505099302472505099603465653002504461653002404486653003504510653002504545653002204570653002204592653003904614653003504653700003604688041001904724989003804743490003704781028003004818856007204848082001104920520160704931EB000712304EBX0100000000000000056538600000000000000.0cr|||||||||||||||||||||140122 ||| eng a97894009383591 aNastasescu, C.00aDimensions of Ring TheoryhElektronische Ressourcecby C. Nastasescu, Freddy Van Oystaeyen a1st ed. 1987 aDordrechtbSpringer Netherlandsc1987, 1987 aXI, 360 pbonline resource0 a1. Finiteness Conditions for Lattices -- 1.1. Lattices -- 1.2. Noetherian and Artinian Lattices -- 1.3. Lattices of Finite Length -- 1.4. Irreducible Elements in a Lattice -- 1.5. Goldie Dimension of a Modular Lattice -- 1.6. Goldie Dimension and Chain Conditions for Modular Lattices with Finite Group Actions -- 1.7. Complements and Pseudo-Complements -- 1.8. Semiatomic Lattices and Compactly Generated Lattices -- 1.9. Semiartinian Lattices -- 1.10. Indecomposable Elements in a Lattice -- 1.11. Exercises -- Bibliographical Comments to Chapter 1 -- 2. Finiteness Conditions for Modules -- 2.1. Modules -- 2.2. The Lattice of Submodules of a Module -- 2.3. Noetherian and Artinian Modules -- 2.4. Modules of Finite Length -- 2.5. Semisimple Modules -- 2.6. Semisimple and Simple Artinian Rings -- 2.7. The Jacobson Radical and the Prime Radical of a Ring -- 2.8. Rings of Fractions. Goldie’s Theorems -- 2.9. Artinian Modules which are Noetherian -- 0 a10.1. Definitions and Basic Properties -- 10.2. GK-dimension of Filtered and Graded Algebras -- 10.3. Applications to Special Classes of Rings -- 10.4. Exercises -- Bibliographical Comments to Chapter 10 -- References0 a2.10. Projective and Infective Modules -- 2.11. Tensor Product and Flat Modules -- 2.12. Normalizing Extensions of a Ring -- 2.13. Graded Rings and Modules -- 2.14. Graded Rings and Modules of Type ?. Internal Homogenisation -- 2.15. Noetherian Modules over Graded Rings of Type ?. Applications -- 2.16. Strongly Graded Rings and Clifford Systems for Finite Groups -- 2.17. Invariants of a Finite Group Action -- 2.18. Exercises -- Bibliographical Comments to Chapter 2 -- 3. Krull Dimension and Gabriel Dimension of an Ordered Set -- 3.1. Definitions and Basic Properties -- 3.2. The Krull Dimension of a Modular Lattice -- 3.3. Critical Composition Series of a Lattice -- 3.4. The Gabriel Dimension of a Modular Lattice -- 3.5. Comparison of Krull and Gabriel Dimension -- 3.6. Exercises -- Bibliographical Comments to Chapter 3 -- 4. Krull Dimension and Gabriel Dimension ofRings and Modules -- 4.1. Definitions and Generalities -- 0 a7.2. The Lattices CF (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 -- 0 a4.2. Krull and Gabriel Dimension of Some Special Classes of Rings and Modules -- 4.3. Exercises -- Bibliographical Comments to Chapter 4 -- 5. Rings with Krull Dimension -- 5.1. Nil Ideals -- 5.2. Semiprime Rings with Krull Dimension -- 5.3. Classical Krull Dimension of a Ring -- 5.4. Associated prime Ideals -- 5.5. Fully Left Bounded Rings with Krull Dimension -- 5.6. Examples of Noetherian Rings of Arbitrary Krull Dimension -- 5.7. Exercises -- Bibliographical Comments to Chapter 5 -- 6. Krull Dimension of Noetherian Rings. The Principal Ideal Theorem -- 6.1. Fully Left Bounded Left Noetherian Rings -- 6.2. The Reduced Rank of a Module -- 6.3. Noetherian Rings Satisfying Condition H -- 6.4. Fully Bounded Noetherian Rings -- 6.5. Krull Dimension and Invertible Ideals in a Noetherian Ring -- 6.6. The Principal Ideal Theorem -- 6.7. Exercises -- Bibliographical Comments to Chapter 6 -- 7. Relative Krull and Gabriel Dimensions -- 7.1. Additive Topologies and Torsion Theories -- aAssociative algebras aCommutative algebra aCommutative Rings and Algebras aNonassociative rings aCommutative rings aAssociative rings aNon-associative Rings and Algebras aAssociative Rings and Algebras1 aVan Oystaeyen, Freddye[author]07aeng2ISO 639-2 bSBAaSpringer Book Archives -20040 aMathematics and Its Applications50a10.1007/978-94-009-3835-940uhttps://doi.org/10.1007/978-94-009-3835-9?nosfx=yxVerlag3Volltext0 a512.44 aApproach 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