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E-raamat: Cyclic Modules and the Structure of Rings

(Professor, Russian State University of Trade and Economics, Moscow, Russia), (Assistant Professor, Saint Louis University, USA), (King Abdulaziz University, SA and Ohio University, USA)
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Cyclic Modules and the Structure of RingsSuurem pilt
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This unique and comprehensive volume provides an up-to-date account of the literature on the subject of determining the structure of rings over which cyclic modules or proper cyclic modules have a finiteness condition or a homological property. The finiteness conditions and homological properties are closely interrelated in the sense that either hypothesis induces the other in some form. This is the first book to bring all of this important material on the subject together.

Over the last 25 years or more numerous mathematicians have investigated rings whose factor rings or factor modules have a finiteness condition or a homological property. They made important contributions leading to new directions and questions, which are listed at the end of each chapter for the benefit of future researchers. There is a wealth of material on the topic which is combined in this book, it contains more than 200 references and is not claimed to be exhaustive.

This book will appeal to graduate students, researchers, and professionals in algebra with a knowledge of basic noncommutative ring theory, as well as module theory and homological algebra, equivalent to a one-year graduate course in the theory of rings and modules.
1 Preliminaries 1(13)
1.1 Artinian and noetherian modules
2(2)
1.2 Free modules, projective modules, and injective modules
4(4)
1.3 Hereditary and semihereditary rings
8(2)
1.4 Generalizations of injectivity
10(4)
2 Rings characterized by their proper factor rings 14(14)
2.1 Restricted artinian rings
14(2)
2.2 Restricted perfect rings
16(3)
2.3 Restricted von Neumann regular rings
19(3)
2.4 Restricted self-injective rings
22(6)
3 Rings each of whose proper cyclic modules has a chain condition 28(9)
3.1 Rings each of whose proper cyclic modules is artinian
28(2)
3.2 Rings with restricted minimum condition
30(4)
3.3 Rings each of whose proper cyclic modules is perfect
34(3)
4 Rings each of whose cyclic modules is injective (or CS) 37(8)
4.1 Rings where each cyclic module is injective
37(2)
4.2 Rings each of whose cyclic modules is CS
39(6)
5 Rings each of whose proper cyclic modules is injective 45(4)
6 Rings each of whose simple modules is injective (or Σ-injective) 49(16)
6.1 V-rings
49(2)
6.2 WV-rings
51(4)
6.3 Σ-V rings
55(8)
6.4 CSI rings
63(2)
7 Rings each of whose (proper) cyclic modules is quasi-injective 65(6)
7.1 Rings each of whose cyclic modules is quasi-injective
65(1)
7.2 Rings each of whose proper cyclic modules is quasi-injective
66(5)
8 Rings each of whose (proper) cyclic modules is continuous 71(4)
8.1 Rings each of whose cyclic modules is continuous
71(1)
8.2 Rings each of whose proper cyclic modules is continuous
72(3)
9 Rings each of whose (proper) cyclic modules is π-injective 75(9)
9.1 Rings each of whose cyclic modules is π-injective
75(4)
9.2 Rings each of whose proper cyclic modules is π-injective
79(5)
10 Rings with cyclics N0-injective, weakly injective, or quasi-projective 84(10)
10.1 Rings each of whose cyclic modules is N0-injective
84(4)
10.2 Rings each of whose cyclic modules is weakly injective
88(2)
10.3 Rings each of whose cyclic modules is quasi-projective
90(4)
11 Hypercyclic, q-hypercyclic, and π-hypercyclic rings 94(19)
11.1 Hypercyclic rings
94(12)
11.2 q-hypercyclic rings
106(4)
11.3 π-hypercyclic rings
110(3)
12 Cyclic modules essentially embeddable in free modules 113(11)
13 Serial and distributive modules 124(7)
14 Rings characterized by decompositions of their cyclic modules 131(16)
15 Rings each of whose modules is a direct sum of cyclic modules 147(4)
16 Rings each of whose modules is an I0-module 151(10)
17 Completely integrally closed modules and rings 161(12)
18 Rings each of whose cyclic modules is completely integrally closed 173(12)
19 Rings characterized by their one-sided ideals 185(22)
19.1 Rings each of whose one-sided ideals is quasi-injective
185(5)
19.2 Rings each of whose one-sided ideals is a direct sum of quasi-injectives
190(10)
19.3 Rings each of whose one-sided ideals is π-injective
200(1)
19.4 Rings each of whose one-sided ideals is a direct sum of π-injective right ideals
200(2)
19.5 Rings each of whose one-sided ideals is weakly injective
202(2)
19.6 Rings each of whose one-sided ideals is quasi-projective
204(3)
References 207(12)
Index 219
S. K. Jain is a Distinguished Professor Emeritus, Ohio University and Advisor, King Abdulaziz University. He was at the Department of Mathematics at Ohio University from 1970-2009. He is an Executive Editor of the Journal of Algebra and its Applications (World Scientific) and Bulletin of Mathematical Sciences (Springer). He is also on the editorial board of the Electronic Journal of Algebra.



Ashish Srivastava is an Assistant Professor of Mathematics at Saint Louis University, Saint Louis, USA. He has written 15 research articles in Noncommutative Algebra and Combinatorics that have been published in various journals.



Askar A. Tuganbaev is a Professor of Mathematics at the Russian State University of Trade and Economics, Moscow, Russia. He has written 10 monographs and more than 180 research articles in Algebra that have been published in various journals.
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