Lectures on Artinian Rings

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