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|a 9789401006521
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|a Puninski, G.
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|a Serial Rings
|h Elektronische Ressource
|c by G. Puninski
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|a 1st ed. 2001
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|a Dordrecht
|b Springer Netherlands
|c 2001, 2001
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|a IX, 226 p
|b online resource
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|a 1 Basic Notions -- 1.1 Preliminaries -- 1.2 Dimensions -- 1.3 Basic ring theory -- 1.4 Serial rings and modules -- 1.5 Ore sets -- 1.6 Semigroup rings -- 2 Finitely Presented Modules over Serial Rings -- 3 Prime Ideals in Serial Rings -- 4 Classical Localizations in Serial Rings -- 5 Serial Rings with the A.C.C. on annihilators and Nonsingular Serial Rings -- 5.1 Serial rings with a.c.c. on annihilators -- 5.2 Nonsingular serial rings -- 6 Serial Prime Goldie Rings -- 7 Noetherian Serial Rings -- 8 Artinian Serial Rings -- 8.1 General theory -- 8.2 d-rings and group rings -- 9 Serial Rings with Krull Dimension -- 10 Model Theory for Modules -- 11 Indecomposable Pure Injective Modules over Serial Rings -- 12 Super-Decomposable Pure Injective Modules over Commutative Valuation Rings -- 13 Pure Injective Modules over Commutative Valuation Domains -- 14 Pure Projective Modules over Nearly Simple Uniserial Domains -- 15 Pure Projective Modules over Exceptional Uniserial Rings -- 16 ?-Pure Injective Modules over Serial Rings -- 17 Endomorphism Rings of Artinian Modules -- Notations
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|a Associative algebras
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| 653 |
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|a Mathematical logic
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|a Algebra
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| 653 |
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|a Associative rings
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| 653 |
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|a Mathematical Logic and Foundations
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| 653 |
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|a Mathematics
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| 653 |
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|a Associative Rings and Algebras
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|a eng
|2 ISO 639-2
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| 989 |
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|b SBA
|a Springer Book Archives -2004
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|a 10.1007/978-94-010-0652-1
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| 856 |
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|u https://doi.org/10.1007/978-94-010-0652-1?nosfx=y
|x Verlag
|3 Volltext
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|a 512.46
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|a The main theme in classical ring theory is the structure theory of rings of a particular kind. For example, no one text book in ring theory could miss the Wedderburn-Artin theorem, which says that a ring R is semisimple Artinian iffR is isomorphic to a finite direct sum of full matrix rings over skew fields. This is an example of a finiteness condition which, at least historically, has dominated in ring theory. Ifwe would like to consider a requirement of a lattice-theoretical type, other than being Artinian or Noetherian, the most natural is uni-seriality. Here a module M is called uni-serial if its lattice of submodules is a chain, and a ring R is uni-serial if both RR and RR are uni-serial modules. The class of uni-serial rings includes commutative valuation rings and closed under homomorphic images. But it is not closed under direct sums nor with respect to Morita equivalence: a matrix ring over a uni-serial ring is not uni-serial. There is a class of rings which is very close to uni-serial but closed under the constructions just mentioned: serial rings. A ring R is called serial if RR and RR is a direct sum (necessarily finite) of uni-serial modules. Amongst others this class includes triangular matrix rings over a skew field. Also if F is a finite field of characteristic p and G is a finite group with a cyclic normal p-Sylow subgroup, then the group ring FG is serial
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