Algebra I by Dr. B. L. van der Waerden (auth.)

By Dr. B. L. van der Waerden (auth.)

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Sample text

Beispiele sind die früher definierten (eineindeutigen) Isomorphismen von Gruppen. h. ist jedes Element von 91 Bildelement mindestens eines a aus IDl, so hat man einen Homomorphismus von m auf 91. Eine homomorphe Abbildung von 9Jl in sich selbst heißt Endamorphismus von m. Bei einer homomorphen Abbildung von 9Jl auf 9Jl kann man die Elemente von m, die ein festes Bild ii in IDl haben, zu einer Klasse a vereinigen. Jedes Element a gehört einer und nur einer Klasse a an; d. h. die Menge 9Jl ist in Klassen eingeteilt, die den Elementen von §R eineindeutig zugeordnet sind.

Nebenklassen Unter einem Komplex versteht man in der Gruppentheorie eine beliebige Menge von Elementen einer Gruppe @. Unter dem Produkt g 9 zweier Komplexe g und 9 versteht man die Menge aller Produkte g h, wo g aus g und h aus 9 entnommen ist. Besteht in dem Produkt g 9der eine Komplex, etwa g, nur aus einem Element g, so schreibt man für g 9 einfach g9. Offenbar gilt die Regel g(9f) = (g9) f. In zusammengesetzten Produkten von Komplexen können die Klammern also weggelassen werden [vgl. § 6, (1)].

Ein System mit doppelter Komposition heißt ein Ring, wenn folgende Rechengesetze für alle Elemente des Systems erfüllt sind: I. Gesetze der Addition. a) Assoziatives Gesetz: a (b c) = (a b) c. b) Kommutatives Gesetz: a b = b a. c) Lösbarkeit! der Gleichung a x = b für alle a und b. II. Gesetz der Multiplikation. a) Assoziatives Gesetz: a · bc =ab· c. 111. Distributivgesetze. a) a · (b + c) =ab+ ac. b) (b+c)·a=ba+ca. + + + + + + + + Zusatz. Gilt auch für die Multiplikation das kommutative Gesetz: II.

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