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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.
List of contents
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.
Summary
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.
