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

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

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Integritätsbereich ist, sobald meiner ist. § 15. Ideale. Restklassenringe Es sei o ein Ring. Damit eine Untermenge von o wieder ein Ring ( Unterring von o) ist, ist notwendig und hinreichend, daß sie I. eine Untergruppe der additiven Gruppe ist, mit anderen Worten zu a undbaucha-b enthält (Moduleigenschaft), 2. zu a und b auch ab enthält. Unter den Unterringen spielen nun einige, die wir Ideale nennen, eine Sonderrolle, analog den Normalteilern in der Gruppentheorie. Eine nichtleere Untermenge m von o heißt Ideal, und zwar Rechtsideal, wenn l.

Was heißt a ""'b(O)? 4. Alle Vielfachen ra eines Elements a bilden ein Linksideal oa. Man mache sich am Ring der geraden Zahlen klar, daß dieses Ideal nicht notwendig mit dem Links-Hauptideal (a) übereinstimmt. Die zweiseitigen Ideale stehen in derselben Beziehung zum Begriff der Ringhornamorphie wie die Normalt,eiler zu dem der Gruppenhomomorphie. Gehen wir vom Hornamorphiebegriff aus! Ein Homomorphismus o --+ definiert eine Klasseneinteilung des Ringes o: eine Klasse Sl'a wird gebildet von allen Elementen a, die o Ideale.

2. Das Einheitsideal o, das alle Elemente des Ringes umfaßt. 49 Ideale. Restklassenringe 3. Das von einem Element a erzeugte Ideal (a), das aus allen Ausdrücken der Gestalt (r E o, n eine ganze Zahl) ra+na besteht. Daß diese Menge stets ein Ideal ist, sieht man leicht ein: Die Differenz zweier solcher Ausdrücke hat offenbar wieder dieselbe Gestalt, und ein beliebiges Vielfaches hat die Form s· (ra +na) = (sr+ ns) ·a, + also die Form r' a oder r' a 0 · a. Das Ideal (a) ist offenbar das kleinste (am wenigsten umfassende) Ideal, das a enthält; denn jedes solche Ideal muß mindestens alle Vielfachen ra und alle Summen ± ,L: a = na enthalten, also auch alle Summen ra + na.

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