Noetherian module

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위키데이터

• ID : Q2444982

말뭉치

1. A right Noetherian ring R is, by definition, a Noetherian right R module over itself using multiplication on the right.
2. Likewise a ring is called left Noetherian ring when R is Noetherian considered as a left R module.
3. The Noetherian condition can also be defined on bimodule structures as well: a Noetherian bimodule is a bimodule whose poset of sub-bimodules satisfies the ascending chain condition.
4. Since a sub-bimodule of an R-S bimodule M is in particular a left R-module, if M considered as a left R module were Noetherian, then M is automatically a Noetherian bimodule.
5. We say that a ring is left (right) Noetherian if it is Noetherian as a left (right) -module.
6. Hilbert's Basis Theorem guarantees that if is a Noetherian ring, then is also a Noetherian ring, for finite .
7. Let be a Noetherian module over a commutative unital ring .
8. First suppose thatis Noetherian.
9. Thenis a submodule of, sois Noetherian.
10. If is a submodule of , then is isomorphic to a submodule of the Noetherian module , so is generated by finitely many elements .
11. The quotient is isomorphic (via ) to a submodule of the Noetherian module , so is generated by finitely many elements .
12. Thus K and Q noetherian implies M is noetherian, and similarly for artinian.
13. A. However, each Tj is its own submodule in M/B. This because each x in Tj represents a unique coset of B. Therefore M/B is not noetherian, and that is a contradiction.
14. as the quotient of a finitely generated free R module F, which is noetherian.
15. A module M which satisfies the two properties in the above theorem is said to be (left) noetherian.
16. Thus a finitely generated semisimple module is noetherian.
17. is a finitely generated Z-module, so it is noetherian as a Z-module.
18. Reversing the direction of inclusion in the definition of noetherian rings, we get a similar concept.
19. M m×n (R) fulfilled Noetherian condition.
20. A characterization of noetherian rings by cyclic modules Proceedings of the Edinburgh Mathematical Society Downloaded from https://www.cambridge.org/core.

메타데이터

위키데이터

• ID : Q2444982

Spacy 패턴 목록

• [{'LOWER': 'noetherian'}, {'LEMMA': 'module'}]