ENGLISH

Lectures on Artinian Rings

By Andor Kertész; Alfred Wiegandt (editor)

Book information

Publisher
Akademiai Kiado
Year
1987
ISBN
9789630543095, 9630543095
Language
english
Format
PDF
Filesize
26 MB (27369429 bytes)
Series
Disquisitiones Mathematica Hungaricae 14
Pages
427\427
Time added
2020-07-14 01:12:09

Description

Preface of the editor From the preface to the German edition Chapter I. Sets, relations 1. Sets, relations, mappings 2. Partially ordered and ordered sets 3. The Kuratowski-Zorn Lemma 4. Abstract dependence Exercises to Chapter I Hints References to Chapter I Chapter II. General properties of rings 5. Rings 6. Ideals, factor rings 7. Rings of power series and rings of polynomials 8. Full matrix rings 9. Embeddings of rings, the Dorroh extension 10. Direct sums of rings 11. Subdirect sums of rings 12. Prime ideals and prime rings 13. Regular rings and their subdirect representations 14. Abelian groups Exercises to Chapter II Hints 2.2-2.14 Hints 2.15-2.52 Hints 2.53-2.81 Hints 2.84-2.85 References to Chapter II Chapter III. Modules and algebras 15. R-modules 16. A module-theoretic characterization of the Dorroh extension 17. Free modules and projective modules 18. Simple modules and completely reducible modules 19. A characterization of completely reducible modules 20. Vector spaces 21. Algebras Exercises to Chapter III Hints References to Chapter III Chapter IV. The radical 22. Primitive rings and primitive ideals, modular right ideals 23. Examples of primitive rings 24. The radical of a ring 25. Some characterizations of the radical 26. The radicals of related rings Exercises to Chapter IV Hints 4.4-4.6 Hints 4.8-4.23 Hints 4.24-4.32 References to Chapter IV Chapter V. Artinian rings in general 27. Artinian and noetherian modules 28. Artinian and noetherian rings 29. Minimum condition and maximum condition for left ideals 30. Nilpotent right ideals. The radical of an artinian ring 31. Non-nilpotent right ideals. Idempotent elements 32. Further results on idempotents 33. The socle of a module and of a ring 34. The radical of an algebra Exercises to Chapter V Hints 5.2-5.12 Hints 5.13-5.24 Hints 5.26-5.29 References to Chapter V Chapter VI. Rings of linear transformations 35. Vector spaces and rings of matrices 36. Left ideals and automorphisms of a matrix ring over a field 37. A Galois connection for finite dimensional vector spaces 38. The Density Theorem of Jacobson 39. The finite topology of Hom_K (V, V) 40. Some consequences of the Density Theorem 41. The Wedderburn-Artin Theorem 42. The Litoff-Ánh Theorem (by R. Wiegandt) 43. Regularity of linear transformations Exercises to Chapter VI Hints 6.2-6.14 References to Chapter VI Chapter VII. Semi-simple, primary and completely primary rings 44. Quasi-ideals 45. Ideal-theoretic characterization of semi-simple rings 46. Maschke's Theorem 47. Indecomposable right ideals and completely primary rings 48. The representation of artinian rings as direct sums of indecomposable right ideals 49. Primary rings Exercises to Chapter VII Hints 7.1-7.15 References to Chapter VII Chapter VIII. Artinian rings as operator domains 50. Semi-simple rings, projective and injective modules 51. Modules over semi-simple rings 52. Systems of equations over modules 53. Injective modules and semi-simple rings 54. Systems of linear equations over semi-simple rings 55. The injective hull (by R. Wiegandt) 56. A characterization of artinian modules (by R. Wiegandt) Exercises to Chapter VIII Hints 8.1-8.6 References to Chapter VIII Chapter IX. The additive groups of artinian rings 57. General remarks on the additive groups of rings 58. The additive groups of artinian rings 59. Artinian rings which are noetherian 60. Noetherian rings which are artinian 61. Artinian rings with identity 62. The splitting of artinian rings 63. Embedding theorems for artinian rings 64. Abelian groups whose full endomorphism rings are artinian Exercises to Chapter IX Hints 9.2-9.15 References to Chapter IX Chapter X. Decomposition of artinian rings (by A. Widiger) 65. Strictly artinian rings 66. The general decomposition theorem 67. Hereditarily artinian rings. Applications Exercises to Chapter X Hints 10.3-10.6 References to Chapter X Chapter XI. Artinian rings of quotients (by G. Betsch) 68. Prerequisites, notations and formulation of the problem 69. The Theorems of Goldie 70. Noetherian orders in artinian rings Exercises to Chapter XI References to Chapter XI Chapter XII. Group rings. A theorem of Connell (by G. Betsch) 71. Group rings 72. Noetherian, regular and semi-simple group rings 73. Artinian group rings Exercises to Chapter XII Hints 12.3-12.6 References to Chapter XII Chapter XIII. Quasi·Frobenius rings (by G. Betsch) 74. Preliminaries 75. The main theorem on QF-rings 76. Modules over QF-rings Exercises to Chapter XIII Hints References to Chapter XIII Chapter XIV. Rings with minimum condition on principal right ideals (by R. Wiegandt) 77. Simple MHR-rings 78. Semi-primitive and radical MHR-rings 79. Rees matrix rings 80. More on MHR-rings 81. The splitting of MHR-rings Exercises to Chapter XIV Hints 14.1-14.4 Hints 14.5-14.7 References to Chapter XIV Chapter XV. Linearly compact rings (by A. Widiger) 82. Topological modules 83. Linearly compact modules and rings 84. Semi-primitive linearly compact rings 85. Decomposition of strictly linearly compact rings into direct sums of right ideals 86. Linearly compact rings whose radicals are linearly compact groups Exercises to Chapter XV Hints References to Chapter XV Hints for the solution of the exercises Bibliography List of symbols Author index Subject index

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