Download e-book for kindle: Assoziative Algebren by Hartmut Laue

By Hartmut Laue

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1 Genau dann ist V ein Zero-Modul, wenn V (1A δ) = {0V } gilt; genau dann unital, wenn V (Aδ) = V gilt. Aus V (1A δ) = {0V } folgt n¨amlich V (xδ) = V (1A x)δ = V (1A δ) (xδ) = {0V } f¨ ur jedes x ∈ A. 1 festgestellten Gleichheit Fix (1A δ) = V (Aδ). Daraus erhalten wir die Folgerung Jeder A-Algebren-Modul hat eine direkte Zerlegung in einen unitalen und einen Zero-Modul. 2 Ist (A; ρ) vollreduzibel und V unital, so ist V vollreduzibel. Ist V unital und irreduzibel, so gilt V ∼ ur ein minimales Rechtsideal R von A.

T¯s β} = {U¯2 , . . , U anderausf¨ uhrung Tj → T¯j → U¯i → Ui ist ein A-Isomorphismus von W auf β ˙ · · · ⊕U ˙ r , der {T2 , . . , Tr } auf {U2 , . . , Ur } abbildet. Also gibt es einen U2 ⊕ A-Automorphismus γ von A mit {T2 γ, . . , Tr γ} = {U2 , . . , Ur }, γ|U1 = idU1 . Nun gilt {T1 αγ, . . , Tr αγ} = {U1 , . . , Ur }, was die Behauptung zeigt. 1: Folgerung Sind D, E Schiefk¨orper, m, n ∈ N und D m×m ∼ = E n×n , so folgt m = n. Sei dazu A := D m×m . 8 gibt es dann minimale Rechtsideale R1 , .

Vn ∈ Xn mit u = v1 + · · · + vn . Sind dann Y1 , . . , Yn ∈ K mit Xi ∈ Yi f¨ ur alle i ∈ n, so gibt es aufgrund der Ketten-Eigenschaft von K unter den Mengen Yi eine, die alle u ¨brigen enth¨alt; diese heiße Y. Da dann X1 , . . , Xn ∈ Y und Y ∈ M gilt, folgt aus (b) zum einen u = 0V . Ist aber u = 0V , so folgt aus (a) zum anderen, daß die oben beliebig gegebene Zerlegung von (u =)0V notwendig die triviale, d. h. die leere Summendarstellung gewesen sein muß; es ist also n = 0. Zusammen folgt: YK ∈ M.

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Assoziative Algebren by Hartmut Laue


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