Noetherian ring

Noun

Meaning

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).

Source: en.wiktionary.org

Related terms

noetherian left-Noetherian right-Noetherian

Examples

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

Close letter words and terms

noetherian induction noetherian integral domain noetherian module noetherianity noetian noetic