Algebra, WS 2009 by Martin Goldstern

By Martin Goldstern

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X1 x2 πy1 y2 . c) π → [0]π = I → π, I → π → [0]π = I (analog zum entsprechenden Beweis f¨ ur Normalteiler). 40 Ist I Ideal von R, dann ist die Faktoralgebra (R/I, +, I, −, ·) ein Ring, genannt der Faktoroder Restklassenring von R modulo I. Die Operationen in R/I sind: (x + I) + (y + I) = (x + y) + I (deckt sich mit der Komplexsumme A + B = {a + b | a ∈ A, b ∈ B}), (x + I)(y + I) = xy + I (deckt sich nicht mit dem Komplexprodukt AB = {ab | a ∈ A, b ∈ B}), −(x + I) = (−x) + I, 0 + I = I ist Nullelement.

Beweis. 5). 2) πf ist Kongruenz: Sei i ∈ I, ni > 0. Wir haben:    a1 πf b1      f (a1 ) = f (b1 )  .. ⇒ ωi∗ f (a1 ) . . f (ani ) = ωi∗ f (b1 ) . . f (bni ) ⇒ .       f (ani ) = f (bni ) ani πf bni 33 ⇒ f (ωi a1 . . ani ) = f (ωi b1 . . ) ⇒ ωi a1 . . ani πf ωi b1 . . bni . Die Eindeutigkeit von g ist trivial: g([a]πf ) = g(ν(a)) = (g ◦ ν)(a) = f (a). 3) g ist ein Homomorphismus: Sei i ∈ I, ni > 0, dann gilt: g(ωi [a1 ]πf . . [ani ]πf ) = g([ωi a1 . . ani ]πf ) = f (ωi a1 .

Die beiden Zuordnungen sind invers: π → [e]π =: N → π1 mit xπ1 y :⇔ x−1 y ∈ N ⇔ xπy, d. , π = π1 . Umkehrung: N → π → [e]π = N . 7 englisch: normal subgroup 35 Um — bis auf Isomorphie — alle homomorphen Bilder einer Gruppe G zu finden, kann man daher alle Normalteiler N von G bestimmen und mit den entsprechenden Kongruenzen die Faktoralgebren G/π bilden. Entspricht dem Normalteiler N die Kongruenz π, so schreiben wir G/N := G/π = {xN | x ∈ G}. Eine solche Faktoralgebra heißt Faktorgruppe von G. In der Faktorgruppe G/N wird wie folgt gerechnet: (xN )(yN ) = (xy)N , eN = N ist Einselement, (xN )−1 = x−1 N .

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