By Walter Borho, Peter Gabriel, Rudolf Rentschler (auth.)

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Algebra. Rings, modules and categories

VI of Oregon lectures in 1962, Bass gave simplified proofs of a few "Morita Theorems", incorporating principles of Chase and Schanuel. one of many Morita theorems characterizes while there's an equivalence of different types mod-A R::! mod-B for 2 jewelry A and B. Morita's answer organizes principles so successfully that the classical Wedderburn-Artin theorem is a straightforward final result, and additionally, a similarity category [AJ within the Brauer team Br(k) of Azumaya algebras over a commutative ring okay comprises all algebras B such that the corresponding different types mod-A and mod-B which include k-linear morphisms are identical through a k-linear functor.

Matrix Partial Orders, Shorted Operators and Applications (Series in Algebra)

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Geometry and Algebra in Ancient Civilizations

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Extra info for Primideale in Einhüllenden auflösbarer Lie-Algebren: Beschreibung durch Bahnenräume

Sample text

Erf~llen sehauen b x~ ~ Eigenvektoren, bleiben, liegt steht). Sei einen (M u @~N 9) Um nicht a' @ b' q = XM@~N bzw. Wir w ~ h l e n mit (x N h - Moduln die a @ b # O. = L ~ bq. ((x M - Yp) @ I + I @ (x N -6q)) m + n ( a p ~ b q ) > I. = dim ~ ~-Eigenvektoren Und P fGr n (x N _ ~q)n+1 denn }-- Schritt: ist = - Yp - 6 q ) m + n ( a p @ bq) i+j=m+n eins # 0 ~ -) E i g e n v e k t o r e n . - in L, L ~(X 2. } f~r ist l&~t. verabreden Element" den 5 5 - s6 R ein ein erzeugt, Quotientenring. s~ R "Lokalisierung Ist Eigenvektor ~ Element, das R, einzigen zu G b e r n e h m e n .

Ist sieh Sei k nicht A @ k K = O. A A @ k K. K' K-rationales auf fur 3. von von ist KSrper. A @ k K = A @ H(H @ kK), Rat selbe zu man zeigen k-Einbettung ein A @ kK---+K ein a p kann zu @ 0 A @ k K' in p~ F dem mit 2. PnA yon = 0, = Q(A) @ k K Fall, under noetherscher A. so Primring. 5). umfaSt Der den P S -I S die P die Durchschnitt Durchschnitt bezeichne rationalen rationalen dieser der P~ S-Ip e-10. 11 hin, c-I0 = 0. 5 einem H(P) sich l£~t. derart A ~ kB zugeordnet, Primidealen Herzen: 0hne ausdehnen dem KSrper ideal Satz Qed.

Stklassenk6rper Yon Sei A@ I Spec A @ k K ---+Spee A. Bi~ektion mit dem S p e k t r u m des "erweiterten Herzens" c) k. A. a) a) P Seien Es folgt p nach I, J I@K, b) J@K C P H([) des Primideals P zu~eordnet ist. Ideale yon und daher WK1p--~* Spec H(P)@kK H(P)@kK. zwei Ideale yon J@K A existiert u n d ist gleich dem H(P) = Q ( H ( P ) @ k K / [ ) = das Dann sind P6~K -I p ~Spec J C P. A mit ASkK IJ C P ~ A , mit Produkt Mit P ~P, abet aber ist also auch prim. b) A/p @k K, Die Primideale die zu betrachten.