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on1373343213 |
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OCoLC |
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20250225213021.0 |
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230318s2023 sz o 000 0 eng d |
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|a 1373336483
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|a 9783031192937
|q electronic book
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|a 3031192931
|q electronic book
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|a 10.1007/978-3-031-19293-7
|2 doi
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|a (OCoLC)1373343213
|z (OCoLC)1373336483
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|a eng
|h fre
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|a QA152.3
|b .B68 2023
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|a PBF
|2 bicssc
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|a MAT002000
|2 bisacsh
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|a HCDD
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|a Bourbaki, Nicolas.
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|a Algebra,
|p Chapter 8 /
|c N. Bourbaki.
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|a [New edition]
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|a Cham :
|b Springer,
|c 2023.
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|a 1 online resource (505 p.)
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|a text
|b txt
|2 rdacontent
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|a computer
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|a online resource
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|a Intro -- To the Reader -- CONTENTS -- INTRODUCTION -- CHAPTER VIII. Semisimple Modules and Rings -- 1. ARTINIAN MODULES AND NOETHERIAN MODULES -- 1. Artinian Modules and Noetherian Modules -- 2. Artinian Rings and Noetherian Rings -- 3. Countermodule -- 4. Polynomials with Coefficients in a Noetherian Ring -- Exercises -- 2. THE STRUCTURE OF MODULES OF FINITE LENGTH -- 1. Local Rings -- 2. Weyr-Fitting Decomposition -- 3. Indecomposable Modules and Primordial Modules -- 4. Semiprimordial Modules -- 5. The Structure of Modules of Finite Length -- Exercises -- 3. SIMPLE MODULES
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|a 1. Simple Modules -- 2. Schur's Lemma -- 3. Maximal Submodules -- 4. Simple Modules over an Artinian Ring -- 5. Classes of Simple Modules -- Exercises -- 4. SEMISIMPLE MODULES -- 1. Semisimple Modules -- 2. The homomorphism sum of homomorphisms -- 3. Some Operations on Modules -- 4. Isotypical Modules -- 5. Description of an Isotypical Module -- 6. Isotypical Components of a Module -- 7. Description of a Semisimple Module -- 8. Multiplicities and Lengths in Semisimple Modules -- Exercises -- 5. COMMUTATION -- 1. The Commutant and Bicommutant of a Module -- 2. Generating Modules
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|a 3. The Bicommutant of a Generating Module -- 4. The Countermodule of a Semisimple Module -- 5. Density Theorem -- 6. Application to Field Theory -- Exercises -- 6. MORITA EQUIVALENCE OF MODULES AND ALGEBRAS -- 1. Commutant and Duality -- 2. Generating Modules and Finitely Generated Projective Modules -- 3. Invertible Bimodules and Morita Equivalence -- 4. The Morita Correspondence of Modules -- 5. Ordered Sets of Submodules -- 6. Other Properties Preserved by the Morita Correspondence -- 7. Morita Equivalence of Algebras -- Exercises -- 7. SIMPLE RINGS -- 1. Simple Rings
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|a 2. Modules over a Simple Ring -- 3. Degrees -- 4. Ideals of Simple Rings -- Exercises -- 8. SEMISIMPLE RINGS -- 1. Semisimple Rings -- 2. Modules over a Semisimple Ring -- 3. Factors of a Semisimple Ring -- 4. Idempotents and Semisimple Rings -- Exercises -- 9. RADICAL -- 1. The Radical of a Module -- 2. The Radical of a Ring -- 3. Nakayama's Lemma -- 4. Lifts of Idempotents -- 5. Projective Cover of a Module -- Exercises -- 10. MODULES OVER AN ARTINIAN RING -- 1. The Radical of an Artinian Ring -- 2. Modules over an Artinian Ring -- 3. Projective Modules over an Artinian Ring -- Exercises
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|a 11. GROTHENDIECK GROUPS Modules -- 1. Additive Functions of Modules -- 2. The Grothendieck Group of an Additive Set of Modules -- 3. Using Composition Series -- 4. The Grothendieck Group R(A) -- 5. Change of Rings -- 6. The Grothendieck Group R(A) -- 7. Multiplicative Structure on K(C) -- 8. The Grothendieck Group K(A) -- 9. The Grothendieck Group K(A) of an Artinian Ring -- 10. Change of Rings for K(A) -- 11. Frobenius Reciprocity -- 12. The Case of Simple Rings -- Exercises -- 12. TENSOR PRODUCTS OF SEMISIMPLE MODULES -- 1. Semisimple Modules over Tensor Products of Algebras
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|a 2. Tensor Products of Simple Modules
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|a This book is an English translation of an entirely revised version of the 1958 edition of the eighth chapter of the book Algebra, the second Book of the Elements of Mathematics. It is devoted to the study of certain classes of rings and of modules, in particular to the notions of Noetherian or Artinian modules and rings, as well as that of radical. This chapter studies Morita equivalence of module and algebras, it describes the structure of semisimple rings. Various Grothendieck groups are defined that play a universal role for module invariants. The chapter also presents two particular cases of algebras over a field. The theory of central simple algebras is discussed in detail; their classification involves the Brauer group, of which several descriptions are given. Finally, the chapter considers group algebras and applies the general theory to representations of finite groups. At the end of the volume, a historical note taken from the previous edition recounts the evolution of many of the developed notions.
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|a Online resource; title from PDF title page (SpringerLink, viewed March 24, 2023).
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|a Algebra.
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7 |
|a algebra.
|2 aat
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| 650 |
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|a Álgebra
|2 embne
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| 650 |
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|a Algebra
|2 fast
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|a Erné, Reinie.
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| 776 |
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|i Print version:
|a Bourbaki, N.
|t Algebra
|d Cham : Springer International Publishing AG,c2023
|z 9783031192920
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| 856 |
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|u https://holycross.idm.oclc.org/login?auth=cas&url=https://link.springer.com/10.1007/978-3-031-19293-7
|y Click for online access
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|a SPRING-MATH2022
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|a 92
|b HCD
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