Некоммутативная лемма Артина-Риса
Sep. 23rd, 2011 05:03 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
posicK. Goodearl, R. Warfield, An introduction to noncommutative Noetherian rings. Cambridge University Press, 2004.
Theorem 13.3 [Artin, Rees]. If R is a noetherian ring and I an ideal of R generated by central elements, then R(I) is noetherian, and hence I satisfies the AR property.
Proof. The noetherian assumption ensures that I can be generated by finitely many central elements, say a1, ..., an. Then R(I) is generated (as a ring) by R together with the central elements a1x, ..., anx. Consequently, R(I) is a homomorphic image of a polynomial ring R[x1,...,xn], and hence it is noetherian by the Hilbert Basis Theorem.
... Как-то я даже не осознавал, что между термином "кольцо Риса" и леммой Артина-Риса есть такая простая прямая связь. Позабыл все, чему учился на первом курсе (когда читал Атью-Макдональда), видимо.
Theorem 13.3 [Artin, Rees]. If R is a noetherian ring and I an ideal of R generated by central elements, then R(I) is noetherian, and hence I satisfies the AR property.
Proof. The noetherian assumption ensures that I can be generated by finitely many central elements, say a1, ..., an. Then R(I) is generated (as a ring) by R together with the central elements a1x, ..., anx. Consequently, R(I) is a homomorphic image of a polynomial ring R[x1,...,xn], and hence it is noetherian by the Hilbert Basis Theorem.
... Как-то я даже не осознавал, что между термином "кольцо Риса" и леммой Артина-Риса есть такая простая прямая связь. Позабыл все, чему учился на первом курсе (когда читал Атью-Макдональда), видимо.
