# Noetherian ring

## English

English Wikipedia has an article on:
Wikipedia

### Etymology

Named after German mathematician Emmy Noether (1882–1935).

### Pronunciation

• IPA(key): /nə.ˈtɛ.ɹi.ən ˈɹɪŋɡ/

### Noun

Noetherian ring (plural Noetherian rings)

1. (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).
• 1986, M. Reid (translator), Hideyuki Matsumura, Commutative Ring Theory, Cambridge University Press, 1989, Paperback Edition, page ix,
The central position occupied by Noetherian rings in commutative ring theory became evident from her[Noether's] work.
• 2000, John C. McConnell, James Christopher Robson, Lance W. Small, Noncommutative Noetherian Rings, 2nd Edition, American Mathematical Society, page 97,
In this chapter the focus moves from semiprime rings to general Noetherian rings, although it does concentrate on prime and semiprime ideals.
• 2004, K. R. Goodearl, Introduction to the Second Edition, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, 2nd Edition, page viii,
During this same period, the explosive growth of the area of quantum groups provided a large new crop of noetherian rings to be analyzed, and thus gave major impetus to research in noetherian ring theory.

#### Usage notes

• Equivalently, a ring that satisfies the ascending chain condition: any chain of left or of right ideals contains only a finite number of distinct elements.
• That is, if ${\displaystyle I_{1}\subseteq \cdots \subseteq I_{k-1}\subseteq I_{k}\subseteq I_{k+1}\subseteq \cdots }$ is such a chain, then there exists an n such that ${\displaystyle I_{n}=I_{n+1}=\cdots .}$
• On classification:
• Noncommutative rings in general, and therefore noncommutative Noetherian rings in particular, are not the subject a field of study distinct from that of commutative rings. Rather, the distinction is between commutative algebra, which deals with commutative rings and related structures, and the more general noncommutative algebra, in which commutativity is not assumed in the structures studied (i.e., the theory potentially applies to both commutative and noncommutative structures).