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2. , V0 = A ⊕ B, B = 0. Then B is a direct sum of simple submodules that are all injective. This implies that V0 is locally injective. 2) W0 is locally projective. , Rad(W0 ) = 0. 2) it follows that ∇(M, W0 ) = Tot(M, W0 ). Since Rad(W0 ) = 0, we have ∇(M, W0 ) = 0, therefore Tot(M, W0 ) = 0. 2) ii) ⇔ iii): Clear. 3 were first proved in the paper [5] without using the notions of locally injective and locally projective. There the modules V0 and W0 were called left– and right–TOTO–modules respectively.

We set In := {1, . . , n}, and consider subsets ∅ = J ⊆ In . Then let eJ be the idempotent of End(M ) with eJ (M ) = Mj , j∈J eJ ( Mi ) = 0. 3. Let M be a non–zero module of finite length. Then the following statements hold. 1) Any idempotent 0 = e ∈ End(M ) is of the form αeJ α−1 for some ∅ = J ⊆ In and some α ∈ Aut(M ). 36 Chapter II. Fundamental notions and properties 2) Let F := {f ∈ End(M ) | ∃ g ∈ End(M ), ∃ ∅ = J ⊆ In : gf = eJ }. Then Tot(End(M )) = End(M ) \ {αF α−1 | α ∈ Aut(M )}. Proof.

This implies f πι = f 1A = f ∈ ∆(A, M ), which was to be shown. Now consider V1 , V2 ∈ Ω and f ∈ Tot(V1 ⊕ V2 , M ). Then it follows that f ι1 ∈ Tot(V1 , M ) = ∆(V1 , M ), f ι2 ∈ Tot(V2 , M ) = ∆(V2 , M ). Since ∆ is an ideal, these imply f ι1 π1 = f e1 , f ι2 π2 = f e2 ∈ ∆(V1 ⊕ V2 , M ) and f e1 + f e2 = f (e1 + e2 ) = f ∈ ∆(V1 ⊕ V2 , M ). Proof for Ψ. Assume W ∈ Ψ, W = A ⊕ B and f ∈ Tot(M, A). Then ιf ∈ Tot(M, W ) = ∇(M, W ). Then it follows that πιf = 1A f = f ∈ ∇(M, A), which was to be proved.

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Algebra I [Lecture notes] by Thomas Keilen


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