- S. Warner, Duke University, Department of Mathematics, Durham, NC, USA
This text brings the reader to the frontiers of current research in topological rings. The exercises illustrate many results and theorems while a comprehensive bibliography is also included.
The book is aimed at those readers acquainted with some very basic point-set topology and algebra, as normally presented in semester courses at the beginning graduate level or even at the advanced undergraduate level. Familiarity with Hausdorff, metric, compact and locally compact spaces and basic properties of continuous functions, also with groups, rings, fields, vector spaces and modules, and with Zorn's Lemma, is also expected.
North-Holland Mathematics Studies
Preface. Topological Rings and Modules. Examples of topological rings. Topological modules, vector spaces, and algebras. Neighbourhoods of zero. Subrings, ideals, submodules, and subgroups. Quotients and projective limits of rings and modules. Metrizability and Completeness. Metrizable groups. Completions of commutative Haussdorf groups. Completions of topological rings and modules. Baire spaces. Summability. Continuity of inversion and adversion. Local Boundedness. Locally bounded modules and rings. Locally retrobounded division rings. Norms and absolute values. Finite-dimensional vector spaces. Topological division rings. Real Valuations. Real valuations and valuation rings. Discrete valuations. Extensions of real valuations. Complete Local Rings. Noetherian modules and rings. Cohen subrings and complete local rings. Complete discretely valued fields. Complete local Noetherian rings. Complete semilocal Noetherian rings. Primitive and Semisimple Rings. Primitive rings. The radical of a ring. Artinian modules and rings. Linear Compactness and Semisimplicity. Linearly compact rings and modules. Linearly compact semisimple rings. Strongly linearly compact modules. Locally linearly compact semisimple rings. Locally compact semisimple rings. Linear Compactness in Rings with Radical. Linear compactness in rings with radical. Lifting idempotents. Locally compact rings. The radical topology. Complete Local Noetherian Rings. The principal ideal theorem. Krull dimension and regular local rings. Complete regular local rings. The Japanese property. Locally Centrally Linearly Compact Rings. Complete discretely valued fields and division rings. Finite-dimensional algebras. Locally centrally linearly compact rings. Historical Notes. Topologies on commutative rings. Locally and linearly compact rings. Category, duality and existence theorems.
Bibliography. Errata. Index of Names. Index of Symbols and Definitions.