Description 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.
Contents 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.
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