Assoziative Algebren by Hartmut Laue PDF

By Hartmut Laue

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Jacobson, bewies in seiner Dissertation 1907 den Hauptsatz im endlich-dimensionalen Fall. 9. 9(2) A gilt: Ri /M(Ri ) ∼ = Rj . = Rj /M(Rj ) ⇔ Ri ∼ A A Daher sind die homogenen Komponenten von (A/N (A); ρ) genau die Moduln i∈k Ri ∼ = Rj A Ri + N (A) /N (A) (j ∈ X), n also isomorph zu den Moduln Rj /M(Rj ) j mit j ∈ X. 12 sind letztere damit isomorph zu den Aln ×n gebren Dj j j mit j ∈ X. 5. 2 als A/N (A)-Modul (also auch als A-Modul) isomorph zu einem minimalen Rechtsideal von A/N (A), also zu genau einem der Moduln Rj /M(Rj ) mit j ∈ X.

Beweis. 1 k¨onnen wir annehmen, daß A Teilalgebra einer unit¨aren assoziativen Algebra A˜ mit I, A A˜ ist. Wir schreiben 1 f¨ ur das Einselement 2 ˜ von A. Sei y := 1−x. Dann gilt: xy = x−x = yx; also sind x, y vertauschbar und xy nilpotent, da Element von I. Sei n ∈ N mit (xy)n = 0A . Wir setzen n−1 e := j=0 2n − 1 2n−1−j j x y. j Dann gilt: e ≡ x2n−1 mod I, da alle Summanden mit j = 0 das Element xy als Faktor enthalten, somit in I liegen. Also gilt auch e ≡ x mod I, und 2n−1 1=1 2n−1 = (x + y) 2n−1 =e+ j=n 2n − 1 2n−1−j j x y , j also n−1 2 e = j=0 n−1 = j=0 2n − 1 2n−1−j j x y 1− j 2n−1 j=n 2n − 1 2n−1−j j x y j 2n − 1 2n−1−j j x y = e, j da xn y j = 0A f¨ ur j ≥ n.

Seien πi : V → Ui , ψj : V → Tj f¨ ur i ∈ r, j ∈ s die Projektionen bez¨ uglich der beiden angegebenen direkten Zerlegungen von V . Es gilt π1 + · · · + πr = idV = ψ1 + · · · + ψs und damit π1 = idV π1 = i∈s ψi π1 ∈ HomA (V, U1 ) ⊆ EndA V. Es folgt: i∈s (ψi π1 )|U1 = i∈s ψi π1 |U1 = π1|U1 = idU1 . Nach Voraussetur das (ψi π1 )|U1 eine Einheit in EndA U1 ist. zung existiert also ein i ∈ s, f¨ O. B. d. A. sei i = 1, also (ψ1 π1 )|U1 ∈ AutA U1 . Dann ist ψ1 |U1 injektiv, U1 ψ1 ∩ Kern π1 = {0V }. Sei W1 := T1 ∩ Kern π1 .

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Assoziative Algebren by Hartmut Laue


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