Download PDF by Hartmut Laue: Assoziative Algebren

By Hartmut Laue

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Rk ∈ X paarweise verschieden, e1 ∈ R1 , . . , ek ∈ Rk mit e = e1 + · · · + ek , so gilt: X = {R1 , . . , Rk }, e1 , . . , ek sind paarweise orthogonale Idempotente, Rj = ej A f¨ ur alle j ∈ k. 37 Beweis. Es gilt: ˙ · · · ⊕R ˙ k ⊇ e1 A⊕ ˙ · · · ⊕e ˙ k A ⊇ (e1 + · · · + ek )A = eA ⊇ eR = R, R ⊇ R1 ⊕ ˙ · · · ⊕R ˙ k , also X = {R1 , . . , Rk }. F¨ damit Rj = ej Af¨ ur alle j ∈ k, R = R1 ⊕ ur jedes j ∈ k gilt ej = eej = (e1 + · · · + ek )ej = e1 ej + · · · + e2j + · · · + ek ej , ∈R1 ∈Rk aufgrund der Eindeutigkeit der Darstellung eines Elementes als Summe von Elementen von Gliedern einer direkten Zerlegung ej = e2j , e1 ej = · · · = ej−1 ej = ej+1 ej = · · · = ek ej = 0A , also ei ej = 0A f¨ ur i = j.

Ur alle i ∈ k, S := Genauer: Seien e1 , . . 4(2), Ri := ei A f¨ Annρ {e1 , . . , ek }. Dann gilt:47 ˙ · · · ⊕M(R ˙ ˙ N (A) = M(R1 )⊕ k )⊕S. Sei X ⊆ k so gew¨ahlt, daß die Rj mit j ∈ X ein Repr¨asentantensystem f¨ ur die A-Isomorphietypen der Rechtsideale R1 , . . , Rk bilden. F¨ ur jedes j ∈ X sei nj die Anzahl der i ∈ k mit Ri ∼ = Rj , und Dj := EndA (Rj /M(Rj ))− . A Dann gilt die Algebren-Isomorphie A/N (A) ∼ = n ×nj Dj j . j∈X Jeder irreduzible A-Algebren-Modul (V ; δ) mit V (Aδ) = {0V } ist zu genau einem der Moduln Rj /M(Rj ) mit j ∈ X A-isomorph.

Sind x ∈ A, y ∈ Annρ (e), so folgt: xy = (xe)y = x(ey) = 0A , also y ∈ Annρ (A). Offensichtlich gilt auch Annρ (A) ⊆ Annρ (e). 2 Proposition Sei R ein linksunit¨ares Rechtsideal einer assoziativen Algebra A. Seien (V ; δV ), (W ; δW ) A-Algebren-Moduln. Dann gibt es zu jedem A-Epimorphismus σ von V auf W und zu jedem A-Homomorphismus ϕ von R in W einen A-Homomorphismus ψ von R in V mit ϕ = ψσ. V ✯ ✟ ✟ ψ ✟✟ ✟✟ σ ❍❍ ❍❍ ❄ ϕ ❍ ❥❄ ❍ R W Beweis. Sei e eine Linkseins von R, w := eϕ, v ∈ V mit vσ = w. Wir setzen ψ : R → V, x → v(xδV ).

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