A Singular Introduction to Commutative Algebra by by Gert-Martin Greuel; Gerhard Pfister

By Gert-Martin Greuel; Gerhard Pfister

This considerably enlarged moment version goals to steer an additional level within the computational revolution in commutative algebra. this is often the 1st handbook/tutorial to greatly take care of SINGULAR. one of the book’s such a lot certain positive aspects is a brand new, thoroughly unified remedy of the worldwide and native theories. one other function of the e-book is its breadth of assurance of theoretical issues within the parts of commutative algebra closest to algebraic geometry, with algorithmic remedies of virtually each subject.

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Additional info for A Singular Introduction to Commutative Algebra by Gert-Martin Greuel (2007-12-04)

Example text

Ist Keine Klasseneinteilung einer Menge M, so bezeichnen wir die K entsprechende Äquivalenzrelation mit 0 K. Ist umgekehrt 0 eine Äquivalenzrelation in M, so schreiben wir M/0 für die Menge der Klassen der 0 entsprechenden Klasseneinteilung. M/0 heißt die Faktormenge von M nach 0. Ist M = I und 0 die durch x o y .. B. ) Die Allrelation und die identische Relation in einer Menge M (siehe Abschnitt 3) sind - wie man sofort sieht - beide Äquivalenzrelationen, genannt die tr ivialen Äqu ivalenzrelat ionen von M.

H Bei dem Beispiel des Preisvergleiches einer Ware in verschiedenen Geschäften bedeutet X I Y, daß die Ware im Geschäft X gleichviel kostet wie im Geschäft Y. Offensichtlich i s t die Relation I hier reflexiv, symmetrisch und transitiv, also eine Äquivalenzrelation. Ist in einer Halbordnung die Relation I eine Äquivalenzrelation, so heißt :5 e ine Pr äferenz r e la t i on , und wird Pr äfer enz ordnung oder sa hwaahe Ordnun g genannt . 58 Bei dem obigen Beispiel handelt es sich also um eine Präferenzrelation .

Hervorgehoben" wird . Wir müssen uns noch überlegen. daß es bei obiger Definition (im Fall n > 0) nicht auf die Wahl der Vertreter ankommen darf (ansonsten würde durch die Vorschrift 0 in M/0 auf Grund einer möglichen Mehrdeutigkeit keine Operation erklärt sein) . Angenommen. • n, Wegen b i 0 a i• und da 0 Kongruenz relation ist. b 2 . · . a 2. a n). h . b 2 ... · . a 2 . an). Wir erha lten also: 0(01. 02. ·· ·. B . M = l und 0 maßen gebildet : ä 1 +ä 2 = a 1 +a 2• 0 m• so werden die Operationen in i/0 m folgendera 1·a 2 a 1a 2• -a -a das Nullelement ist U und das Einselement T.

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