Noun
Noetherian ring (plural Noetherian rings)
(algebra, ring theory) A ring which is either: (a) a commutative ring in which every ideal is finitely generated, or (b) a noncommutative ring that is both left-Noetherian (every left ideal is finitely generated) and right-Noetherian (every right ideal is finitely generated).
If is a Noetherian ring, then is a Noetherian ring. Source: Internet
In a Noetherian ring, every prime ideal has finite height. Source: Internet
The notion of a Noetherian ring is of fundamental importance in both commutative and noncommutative ring theory, due to the role it plays in simplifying the ideal structure of a ring. Source: Internet
Nonetheless, Nagata gave an example of a Noetherian ring of infinite Krull dimension. Source: Internet
Stated differently, the image of any surjective ring homomorphism of a Noetherian ring is Noetherian. Source: Internet
The integers, however, form a Noetherian ring which is not Artinian. Source: Internet