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By Prof. Dr. Otto Forster (auth.)

Dr. Otto Forster ist Professor am Mathematischen Institut der Ludwig-Maximilians-Universität München und Autor der bekannten Lehrbücher research 1-3.

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Extra info for Algorithmische Zahlentheorie

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Es ist offenbar das kleinste Ideal von R, das x enthalt und heiBt das von x erzeugte Hauptideal. Es wird auch kurz mit (x) bezeichnet, wenn klar ist, welcher Ring zugrunde liegt. b) Etwas allgemeiner seien xl, ... , xr E R. Dann ist ebenfalls ein Ideal, das von Xl, ... , xr erzeugte Ideal. Es wird auch kurz mit (Xl, ... , xr) bezeichnet. 7. Satz. Sei Rein Integritiitsbereich. i) Fur x, y E R gilt x Iy ¢::=} (y) C (x). ii) Zwei Elemente x, y E R " {O} sind genau dann assoziiert, falls (x) = (y).

Satz. Seien x, y zwei von 0 verschiedene Elemente eines Integritiitsbereichs R. Gilt x I y und y I x, so sind x und y assoziiert. Beweis. Nach Voraussetzung gilt y = qlx und x = q2Y mit Elementen qI,q2 E R. Daraus folgt x = qlq2x, also (qlq2 - l)x = o. Da x =f. h. d. 2. Definition. Seien x, y zwei Elemente eines Integritatsbereichs R. Ein Element dE R heiBt gra/3ter gemeinsamer Teiler von x und y, falls folgende beiden Bedingungen erfullt sind: i) d I x und d I y. ii) 1st d' E Rein weiteres Element mit d' I x und d' I y, so folgt d' I d.

Seien x, y zwei Elemente eines Integritatsbereichs R. Ein Element v E R heifit kleinstes gemeinsames Vielfaches von x und y, wenn gilt: i) x I v und y I V. ii) 1st w E Rein weiteres Element mit x Iw und y I w, so folgt v I W. 31 Der euklidische Algorithmus Wie im Fall des gr6f3ten gemeinsamen Teilers zeigt man, dass im Faile der Existenz das kleinste gemeinsame Vielfache bis auf Einheiten eindeutig bestimmt ist. 12. Satz. Sei Rein Hauptidealring und seien x, y E R. Dann existiert ein kleinstes gemeinsames Viel/aches von x und y.

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