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By Hartmut Laue

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Jk von A mit x ∈ J1 + · · · + Jk . Die Algebren (J1 + · · · + Jk )/(J1 + · · · + Jk−1 ), . . , (J1 + J2 )/J1 , J1 sind isomorph zu Faktoralgebren von J1 + · · · + Jk , . . 1(1)). 1(1)), somit x nilpotent. Damit ist N (A) nil. 1(1)), also J ⊆ N (A). Es folgt: N (A/N (A)) = {0A/N (A) }. 3(2) ein nilpotentes Ideal I von A mit R ⊆ I. Da I insbesondere nil ist, folgt R ⊆ I ⊆ N (A). Beispiel. Sei K ein K¨orper und Tn die K-Teilalgebra von K n×n , die aus 44 den unteren Dreiecksmatrizen u ¨ber K besteht.

Ek ∈ A gibt ˙ · · · ⊕e ˙ k A⊕S, ˙ mit A = e1 A⊕ S = Annρ {e1 , . . , ek }. Indem wir k := 0 setzen, erhalten wir: A ∈ M, also M = ∅. 1 ein minimales Element S ∗ . Wir behaupten, daß S ∗ kein Idempotent = 0A enth¨alt und nehmen das Gegenteil an. 2 ein linksunit¨ares Rechtsideal = {0A } von A. 1) in der Menge der linksunit¨aren Rechtsideale = {0A } von A, die in S ∗ liegen, ein minimales Element R. 4). 1 nichttriviale direkte Summe linksunit¨arer Rechtsideale von A w¨are, ein Widerspruch zur Minimalit¨at von R.

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