Modules over endomorphism rings:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2010
|
| Edition: | 1. publ. |
| Series: | Encyclopedia of mathematics and its applications
130 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XX, 372 S. graph. Darst. |
| ISBN: | 9780521199605 |
Staff View
MARC
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| 245 | 1 | 0 | |a Modules over endomorphism rings |c Theodore G. Faticoni |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2010 | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface
page
xiii
1
Preliminary results
1
1.1
Rings, modules, and functors
1
1.2
Azumaya-Krull-Schmidt theorem
3
1.3
The structure of rings
4
1.4
The Arnold-Lady theorem
5
2
Class number of an abelian group
9
2.1
Preliminaries
9
2.2
A functorial bijection
14
2.3
Internal cancellation
17
2.4
Power cancellation
19
2.5
Unique decomposition
21
2.6
Algebraic number fields
23
2.7
Exercises
25
2.8
Problems for future research
25
3
Mayer-Vietoris sequences
26
3.1
The sequence of groups
26
3.2
Analytic methods
32
3.3
Exercises
36
3.4
Problems for future research
36
4
Lifting units
37
4.1
Units and sequences
37
4.2
Calculations with primary ideals
40
4.3
Quadratic number fields
44
4.4
The Gaussian integers
45
4.5
Imaginary quadratic number fields
46
4.6
Exercises
48
4.7
Problems for future research
48
5
The conductor
49
5.1
Introduction
49
5.2
Some functors
51
5.3
Conductor of an rtffr ring
53
5.4
Local correspondence
54
viii Contents
60
60
61
61
63
67
70
72
75
76
77
77
79
81
90
91
93
95
96
98
100
102
102
£-groups
ЮЗ
8.1 J -groups, £-groups,
and
«S-groups 103
8.2 Eichler
groups
103
8.3
Direct sums of jC-groups
106
8.4
Eichler ¿-groups are vZ-groups
109
8.5
Exercises
1П
8.6
Problems for future research
112
Modules and homotopy classes
113
9.1
Right endomorphism modules
113
9.1.1
The homotopy of G-plexes
113
9.1.2
Homotopy and homology
118
9.1.3
Endomorphism modules as G-plexes
120
9.2
Two commutative triangles
128
9.2.1
G-Solvable
Я
-modules
129
9.2.2
A factorization of the tensor functor
131
9.3
Left endomorphism modules
134
9.3.1
Duality
137
9.4
Self-small self-slender modules
142
9.5
(μ)
Implies slender injectives
143
9.6
Exercises
144
9.7
Problems for future research
146
5.5
Exercises
5.6
Problems for future research
Conductors and groups
6.1
Rtffr groups
6.2
Direct sum decompositions
6.3
Locally semi-perfect rings
6.4
Balanced semi-primary groups
6.5
Examples
6.6
Exercises
6.7
Problems for future research
Invertible fractional ideals
7.1
Introduction
7.2
Functors and bijections
7.3
The square
7.4
Isomorphism classes
7.5
The equivalence class
(/}
7.6
Commutative domains
7.7
Cardinality of the kernels
7.8
Relatively prime to
τ
7.9
Power cancellation
7.10
Algebraic number fields
7.11
Exercises
7.12
Problems for future research
Contents
10
11
12
13
Tensor
functor
equivalences
148
10.1
Small
projective
generators
148
10.2
Quasi-projective modules
153
10.3
Flat endomorphism modules
157
10.3.1
A category equivalence for
submodules
of
free modules
157
10.3.2
Right ideals in endomorphism rings
161
10.3.3
A criterion for f-flatness
162
10.4
Orsatti and Menini s *-modules
163
10.5
Dualities from injective properties
166
10.5.1
G-Cosolvable «-modules
167
10.5.2
A factorization of Hom^f-, G)
168
10.5.3
Dualities for the dual functor
169
10.6
Exercises
171
10.7
Problems for future research
173
Characterizing endomorphisms
175
11.1
Flat endomorphism modules
175
11.2
Homological dimension
177
11.2.1
Definitions and examples
177
11.2.2
The exact dimension of a G-plex
179
11.2.3
The
projective
dimension of a G-plex
180
11.3
The flat dimension
184
11.4
Global dimensions
188
11.5
Small global dimensions
191
11.5.1
Baer s lemma
191
11.5.2
Semi-simple rings
194
11.5.3
Right hereditary rings
195
11.5.4
Global dimension at most
3
201
11.6
Injective dimensions and modules
202
11.6.1
A review of G-coplexes
202
11.6.2
Injective endomorphism rings
206
11.6.3
Left homological dimensions
209
11.7
A glossary of terms
212
11.8
Exercises
214
11.9
Problems for future research
218
Projective
modules
219
12.1
Projectives
219
12.2
Finitely generated modules
223
12.3
Exercises
228
12.4
Problems for future research
228
Finitely generated modules
229
13.1
Beaumont-Pierce
229
13.2
Noetherian modules
235
13.3
Generators
237
x
Contents
13.4
Exercises
241
13.5 Problems
for future
research
241
14
Rtffr
E-profeti ve
groups
242
14.1
Introduction
242
14.2
The UConn
81
Theorem
245
14.3
Exercises
248
14.4
Problems for future research
248
15
Injective endomorphism modules
249
249
251
258
261
262
263
264
16
A diagram of categories
265
265
269
272
272
274
275
275
276
279
282
282
17
Diagrams of abelian groups
284
285
286
288
291
294
294
296
298
300
301
304
309
311
18
Marginal
isomorohisrns 3J2
312
313
315
15.1
G-Monomorpmsms
15.2
Injective properties
15.3
G-Cogenerators
15.4
Projective
modules revisited
15.5
Examples
15.6
Exercises
15.7
Problems for future research
A diagram of categories
16.1
The diagram
16.2
Smallness and slenderness
16.3
Coherent objects
16.4
The construction function
16.5
The Greek maps
16.6
Applications
16.6.1
Complete sets of invariants
16.6.2
Unique topological decompositions
16.6.3
Homological dimensions
16.7
Exercises
16.8
Problems for future research
Diagrams of abelian groups
17.1
The ring EndciJf
)
17.2
Topological complexes
17.3
Categories of complexes
17.4
Commutative triangles
17.5
Three diamonds
17.5.1
A diagram for an abelian groups
17.5.2
Self-small and self-slender
17.5.3
Coherent complexes
17.6
Prism diagrams
17.7
Direct sums
17.8
Algebraic number fields
17.9
Exercises
17.10
Problems for future research
Marginal isomorphisms
18.1
Ore localization
18.1.1
Preliminary concepts and examples
18.1.2
Noncommutative
localization
Contents xi
18.2
Marginal
isomorphisms
322
18.2.1
Margimorphism and localizations
323
18.2.2
Marginal summands
327
18.2.3
Marginal summands as projectives
329
18.2.4
Projective QG
-modules
331
18.3
Uniqueness of direct summands
333
18.3.1
Totally indecomposable modules
333
18.3.2
Morphisms of totally
indécomposables
334
18.3.3
Semi-simple marginal summands
337
18.3.4
Jónsson s
theorem and margimorphisms
339
18.4 Nilpotent
sets and margimorphism
342
18.5
Isomorphism from margimorphism
346
18.6
Semi-simple endomorphism rings
353
18.7
Exercises
358
18.8
Problems for future research
360
Bibliography
362
Index
368
MODULES
OVER
ENDOMORPHISM RINGS
This is an
extensive
synthesis of recent work in the study of endomorphism rings
and their modules, bringing together direct sum decompositions of modules, the
class number of an algebraic number field, point set topological spaces, and
classical
noncommutative
localization.
The main idea behind the book is to study modules
G
over a ring
R
via their
endomorphism ring
Endrç(G).
The author discusses a wealth of results that classify
G
and Endn(G) via numerous properties, and in particular results from point set
topology are used to provide a complete characterization of the direct sum
decomposition properties of G.
For graduate students this is a useful introduction, while the more experienced
mathematician will discover that the book contains results that are not otherwise
available. Each chapter contains a list of exercises and problems for future research,
which provide a springboard for students entering modern professional mathematics.
Theodore G.
Faticoni
is Professor in the Mathematics Department at Fordham
University, New York.
|
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| indexdate | 2024-07-09T22:05:04Z |
| institution | BVB |
| isbn | 9780521199605 |
| language | English |
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| series2 | Encyclopedia of mathematics and its applications |
| spelling | Faticoni, Theodore G. Verfasser aut Modules over endomorphism rings Theodore G. Faticoni 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2010 XX, 372 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Encyclopedia of mathematics and its applications 130 Endomorphism rings Modules (Algebra) Endomorphismus (DE-588)4280121-7 gnd rswk-swf Modul (DE-588)4129770-2 gnd rswk-swf Endomorphismus (DE-588)4280121-7 s Modul (DE-588)4129770-2 s DE-604 Encyclopedia of mathematics and its applications 130 (DE-604)BV000903719 130 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018667524&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018667524&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Faticoni, Theodore G. Modules over endomorphism rings Encyclopedia of mathematics and its applications Endomorphism rings Modules (Algebra) Endomorphismus (DE-588)4280121-7 gnd Modul (DE-588)4129770-2 gnd |
| subject_GND | (DE-588)4280121-7 (DE-588)4129770-2 |
| title | Modules over endomorphism rings |
| title_auth | Modules over endomorphism rings |
| title_exact_search | Modules over endomorphism rings |
| title_full | Modules over endomorphism rings Theodore G. Faticoni |
| title_fullStr | Modules over endomorphism rings Theodore G. Faticoni |
| title_full_unstemmed | Modules over endomorphism rings Theodore G. Faticoni |
| title_short | Modules over endomorphism rings |
| title_sort | modules over endomorphism rings |
| topic | Endomorphism rings Modules (Algebra) Endomorphismus (DE-588)4280121-7 gnd Modul (DE-588)4129770-2 gnd |
| topic_facet | Endomorphism rings Modules (Algebra) Endomorphismus Modul |
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