Topological Rings Satisfying Compactness ConditionsIntroduction In the last few years a few monographs dedicated to the theory of topolog ical rings have appeared [Warn27], [Warn26], [Wies 19], [Wies 20], [ArnGM]. Ring theory can be viewed as a particular case of Z-algebras. Many general results true for rings can be extended to algebras over commutative rings. In topological algebra the structure theory for two classes of topological algebras is well developed: Banach algebras; and locally compact rings. The theory of Banach algebras uses results of Banach spaces, and the theory of locally compact rings uses the theory of LCA groups. As far as the author knows, the first papers on the theory of locally compact rings were [Pontr1]' [J1], [J2], [JT], [An], lOt], [K1]' [K2]' [K3], [K4], [K5], [K6]. Later two papers, [GS1,GS2]appeared, which contain many results concerning locally compact rings. This book can be used in two w.ays. It contains all necessary elementary results from the theory of topological groups and rings. In order to read these parts of the book the reader needs to know only elementary facts from the theories of groups, rings, modules, topology. The book consists of two parts. |
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additive group affirm algebraic Assume the contrary base consisting canonical homomorphism Cauchy filter closed subgroup cofinite commutative compact group compact right topological compact topology Consider contains continuous homomorphism contradiction Corollary Denote dense direct sum discrete division ring exists a neighborhood finite subset follows fundamental system G ft G N+ group G group topology hence idempotent invariant subgroup Jacobson radical LCA group left ideal Lemma Let G linearly compact locally compact ring locally topologically nilpotent mapping module natural number neighborhood of zero nilpotent ideal nilring non-discrete non-zero obtain Obviously open ideal open subgroup open subset prime number Proof proved quotient ring R-module right ideal right topological ring ring topology ring with identity semisimple submodule subring subspace system of neighborhoods Theorem topological Abelian group topological direct topological group topological product topological ring topological space topologically isomorphic topologically locally finite totally bounded totally disconnected two-sided ideal
Popularni odlomci
Stranica 46 - Y be a continuous mapping of a topological space X into a topological space Y . Then for every transfinite number af(Qx,a) Ç Qf(x),aPROOF.
Stranica 2 - ... and open in y. We shall also consider many-valued mappings. The many-valued mapping /: X —» y maps each point x € X onto a closed set fx of the space y. The many-Valued mapping /: X —» y is said to be continuous (see [ 12 ]) if for every x €. X and every neighborhood Ufx of the set fxCY there exists a neighborhood Ux of the point x such that fUx C Ufx.
Stranica 2 - Y to be open if it is a union of sets of the form U x V, where U is an open subset of X and V is an open subset of У; that is, the collection С of these subsets is a basis for the topology.
Stranica 186 - S is precompact if and only if for each e > 0 there exists a neighborhood U of QG such that S(U) = {s(u) : s € 5, u £ U} Ç <p(Oe).
Stranica 67 - V be a neighborhood of e of G. There exists a finite subset F of G such that G = FHV Since Я satisfies (*) there exists a finite subset K of Я such that Я С K V.
Stranica vii - A noteworthy feature is the extent to which compactness serves as a substitute for the classical chain conditions...
Stranica ix - The additive group of a ring R will be denoted by R(+) and its multiplicative groupoid by R(-).
Stranica 224 - If A is a locally compact ring, A = S + I , where I is a closed locally topologically nilpotent two-sided ideal and S a compact quasiregular subring, then A is locally topologically nilpotent . PROOF.
Bibliografski podaci
| Naslov | Topological Rings Satisfying Compactness Conditions Opseg 549 iz Mathematics and Its Applications |
| Autor | M. Ursul |
| Izdanje | ilustrirano |
| Izdavač | Springer Science & Business Media, 2002 |
| ISBN | 1402009399, 9781402009396 |
| Duljina | Broj stranica: 327 |
| Izvezi navode | BiBTeX EndNote RefMan |