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" Serial Rings "
by Gennadi Puninski.
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BL
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| Record Number
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579499
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b408718
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| Main Entry
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Puninski, Gennadi.
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| Title & Author
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Serial Rings\ by Gennadi Puninski.
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| Publication Statement
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Dordrecht :: Springer Netherlands :: Imprint: Springer,, 2001.
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| ISBN
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9789401006521
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: 9789401038621
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| Contents
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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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| Abstract
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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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Mathematics.
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Algebra.
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Logic, Symbolic and mathematical.
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| Added Entry
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SpringerLink (Online service)
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