Serial Rings Softcover Reprint Of The Original 1st Ed. 2001

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. Front Matter....Pages i-ix Basic Notions....Pages 1-19 Finitely Presented Modules over Serial Rings....Pages 20-33 Prime Ideals in Serial Rings....Pages 34-40 Classical Localizations in Serial Rings....Pages 41-54 Serial Rings with the A.C.C. on annihilators and Nonsingular Serial Rings....Pages 55-63 Serial Prime Goldie Rings....Pages 64-79 Noetherian Serial Rings....Pages 80-92 Artinian Serial Rings....Pages 93-112 Serial Rings with Krull Dimension....Pages 113-122 Model Theory for Modules....Pages 123-135 Indecomposable Pure Injective Modules over Serial Rings....Pages 136-150 Super-Decomposable Pure Injective Modules over Commutative Valuation Rings....Pages 151-162 Pure Injective Modules over Commutative Valuation Domains....Pages 163-171 Pure Projective Modules over Nearly Simple Uniserial Domains....Pages 172-186 Pure Projective Modules over Exceptional Uniserial Rings....Pages 187-197 Σ-Pure Injective Modules over Serial Rings....Pages 198-205 Endomorphism Rings of Artinian Modules....Pages 206-210 Back Matter....Pages 211-226 This book presents an exhaustive and up-to-date overview of the structure theory of serial rings, and the various methods of treating them. Results have been scattered throughout the literature, and the achievements of some schools, such as the Kiev school, seem little-known. This volume endeavours to unify the wide spectrum of tools used in this area and state the theory of serial rings based on two constructions: firstly, localisation with respect to a semi-prime Goldie ideal; and, secondly, a hidden 'blow-up' construction in a serial ring. Part of the work deals with the theory of modules over a serial ring, especially with finitely presented and pure injective modules. Other topics include noetherian serial rings and Artinian serial rings. Audience: This volume can be used as a textbook in ring theory and in the model theory of modules, and will also be of interest to postgraduates and researchers whose work involves rings and algebras.