a ∈ R a = e + u u R R M End(M R ) M A A = M  ⊕ N = ⊕ i∈I A i M  ∼ = M A  i ⊆ A i A  i ⊆ ⊕ A i A = M  ⊕ ( ⊕ i∈I A  i ) M R R R R M R End(M R ) M R C 1 M M C 2 A B M A M B C 3 A B M A ∩ B = 0 A ⊕ B M M M C 1 C 1 M C 1 M C 1 C 2 M M C 1 C 3 M ⇒ ⇒ ⇒ ⇒ ⇒ C 1 A ⊆ M, A ⊆ ⊕ M End(M R ) A M M M M M M u − dim(M) < ∞ M A ⊆ M f ∈ End(M R ) f(A) ⊆ A e R eRe (1 − e)R(1 − e) R M = M 1 ⊕ M 2 ⊕ ⊕ M n M i M M R = M 1 ⊕ M 2 ⊕ ⊕ M n {e 1, e 2 , , e n } End(M R ) e 1 + e 2 + + e n = 1 M M i = Me i ∀i = 1, , n e i End(M R )e i ∼ = End(Me i ). M i End(Me i ) e i End(M R )e i ∀i = 1, , n End(M R ) M R R R / J(R) R J(R) R M R M R X = M ⊕ M X A 1 = M ⊕ 0 , A 2 = 0 ⊕ M. K = {(x, x), x ∈ M}. f, g ∈ End(M R ) f + g = 1 M M  =  (f(x), −g(x)), x ∈ M  M  ∼ = M X = M  ⊕ K (x, y) ∈ X (x, y) =  f(x − y), −g(x − y)  +  f(y) + g(x), f(y) + g(x)  ∈ M  + K. X ⊆ M  + K M  ∩ K = 0 X = M  ⊕ K A  i ⊆ A i X = M  ⊕ A  1 ⊕ A  2 x ∈ M (x, x) =  f(y), −g(y)  + (x 1 , 0) + (0, x 2 ). f  , g  ∈ End(M) f  (x) = x 2 g  (x) = x 1  ff  (x), ff  (x)  =  ff  (x), −gf  (x)  + (0, 0) +  0, f  (x)  .  g.g  (x), g.g  (x)  =  − fg  (x), gg  (x)  +  g  (x), 0  + (0, 0).  ff  (x), ff  (x)  =  f(y 1 ), −g(y 1 )  +  g  ff  (x), 0  +  0, f  ff  (x)  . f  ff  (x) = f  (x) f  ff  = f  g  gg  = g  ff  , gg  S = End(M R ) x = f  (x) − g(y) x = f(y) + g  (x) y = (f  − g  )(x) x = f(y)+g  (x) = (ff  +gg  )(x) ff  +gg  = 1 M f ∈ End(M) e 2 = e = ff  ∈ fS 1 − e ∈ (1 − f)S S = End(M R ) f, e ∈ End(M ) e 2 = e f(M) = Imf ⊆ ⊕ M Im(1−f ) = Kerf ⊆ ⊕ M e  f(M)  ⊆ f(M ) e  (1−f )(M)  ⊆ (1−f)(M) (ef)(M) ⊆ Ker(1−f) Imf = Ker(1−f) (1−f )  ef(M)  = 0 fef = ef (1 − f)e(1 − f) = e(1 − f) e − ef = e(1 − f) = (1 − f)e(1 − f) = e − ef − fe + fef = e − ef − fe + ef = e − fe. ef = fe e S f  f = f  (ff  )f = f  f(ff  ) = (f  f)ff  = f(f  f)f  = f(f  ff  ) = ff  . g  g = gg  u = (f  − g  ) f − (1 − e) S f = (1 − e) + u S M R M R u − dimM < ∞ M R M u−dimM < ∞ M = n ⊕ i=1 U i U i U i ⊆ M M U i M End(U i ) End(U i ) α, β ∈ End(U i ) (α − β) End(U i ) N = N 1 ⊕ N 2 N 1 = (U i , 0) ∼ = U i N 2 = (0, U i ) ∼ = U i f := (α, β) U i N f(x) = (α(x), β(x)) R g ∈ End(U i ) g(x 1 , x 2 ) = (α − β) −1 (x 1 − x 2 ) gf = id U i f N = Imf ⊕ K = N 1 ⊕ N 2 K ⊆ ⊕ N Imf ∼ = U i N  1 ⊆ ⊕ N 1 , N  2 ⊆ ⊕ N 2 N = Im f ⊕ N  1 ⊕ N  2 N 1 = N  1 ⊕ N  1 , N 2 = N  2 ⊕ N  2 Imf ∼ = N / N  1 ⊕ N  2 ∼ = N  1 ⊕ N  2 Imf ∼ = U i N  1 = 0 N  2 = 0 N  1 = 0 N  2 = 0 N = Imf ⊕ N 1 ⊕ N  2 N = N 1 ⊕N 2 N  2  N 2 N 2 N  2 = 0 N = Imf ⊕N 1 π 2 : N → N 2 N N 2 π 2 | Imf : Imf → N 2 β = π 2 f : U i → N 2 ∼ = U i  β End(U i )  End(U i ) End(U i ) U i M = n ⊕ i=1 U i (1−C 1 ) u−dim(M) < ∞ M (1 − C 1 ) u − dim(M) < ∞ M u−dim( M / SocM ) < ∞ M = M 1 ⊕ M 2 M 1 u − dim(M 2 ) < ∞ M 2 M