Download e-book for iPad: Algebra. Rings, modules and categories by Carl Faith

By Carl Faith

ISBN-10: 3540055517

ISBN-13: 9783540055518

VI of Oregon lectures in 1962, Bass gave simplified proofs of a couple of "Morita Theorems", incorporating principles of Chase and Schanuel. one of many Morita theorems characterizes whilst there's an equivalence of different types mod-A R::! mod-B for 2 jewelry A and B. Morita's resolution organizes principles so successfully that the classical Wedderburn-Artin theorem is an easy outcome, and in addition, a similarity category [AJ within the Brauer crew Br(k) of Azumaya algebras over a commutative ring ok involves all algebras B such that the corresponding different types mod-A and mod-B such as k-linear morphisms are identical by way of a k-linear functor. (For fields, Br(k) contains similarity periods of easy primary algebras, and for arbitrary commutative okay, this can be subsumed lower than the Azumaya [51]1 and Auslander-Goldman [60J Brauer crew. ) a variety of different cases of a marriage of ring thought and classification (albeit a shot­ gun wedding!) are inside the textual content. additionally, in. my try and additional simplify proofs, significantly to do away with the necessity for tensor items in Bass's exposition, I exposed a vein of rules and new theorems mendacity wholely inside of ring conception. This constitutes a lot of bankruptcy four -the Morita theorem is Theorem four. 29-and the foundation for it's a corre­ spondence theorem for projective modules (Theorem four. 7) steered by way of the Morita context. As a derivative, this offers origin for a slightly whole idea of easy Noetherian rings-but extra approximately this within the advent.

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Example text

Behauptung: Es sei R ′ ein Unterring des Integrit¨atsbereiches S ′ derart, daß R ′ ein Jacobson-Ring ist und S ′ = R ′ [t] mit einem t ∈ S ′ . Ist Sb′ ein K¨orper f¨ur ein b ∈ S ′ , b ̸ = 0, dann sind R ′ und S ′ K¨orper, und S ′ ⊇ R ′ ist eine endliche K¨orpererweiterung. 5 (mit S an Stelle von R) erf¨ullt. S ist also ein Jacobson-Ring. Ist ferner Q ein maximales Ideal in S, dann ist (S/Q)1 = S/Q ein K¨orper und damit nach der Behauptung auch R/α −1 (Q) ein K¨orper (insbesondere ist Q ∩ R = α −1 (Q) ein maximales Ideal in R), und es gilt [S/Q : R/(Q ∩ R)] < ∞.

Ist K algebraisch abgeschlossen und I ein Ideal in S = K [X 1 , . . , X n ], dann ist rad I = J (V (I )). Beweis. Q sei ein maximales Ideal in S. 1 ist dann Q = Q a = ∑ n ur ein a = (a1 , . . , an ) ∈ K n . F¨ur ein beliebiges von S veri=1 S(X i − ai ) f¨ schiedenes Ideal I in S gilt wegen V (Q a ) = {a} und J ({a}) = Q a Qa ⊇ I ⇐⇒ a ∈ V (I ) ⇐⇒ Q a ⊇ J (V (I )). (∗) Weiter hat man rad I = ∩ P P Primideal, P⊇I = ∩ ( P Primideal, P⊇I = ∩ Q ) ∩ Q (da S ein Jacobson-Ring ist) Q max. , Q⊇P ⊇ J (V (I )), Q max.

N = 0. (∗) x x n−1 + . . + a0 das Minimalpolynom von t u ¨ ber K (da In der Tat ist X n + an−1 x X xn umgekehrt aus einer Gleichung t r + br −1 t r −1 + . . + b0 = 0 mit b j ∈ K durch Multiplikation mit (y/t)r die Gleichung y r + br −1 x y r −1 + . . + b0 x r = 0 folgt). 25), also x an−i ∈ R f¨ur i = 0, . . , n − 1. ai i Angenommen x ̸∈ P0 . Dann folgt aus x n−i · x n−i = ai ∈ P0 , daß x an−i ∈ P0 f¨ur n n i = 0, . . , n − 1 und damit t ∈ P0 S wegen (∗), also auch t ∈ Q 1 . Das ist aber nicht m¨oglich.

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Algebra. Rings, modules and categories by Carl Faith

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