Noetherian ring

English

Alternative forms

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Etymology

Named after German mathematician Template:W (1882–1935).

Pronunciation

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Noun

1. Template:Lb 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).
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   • 2004, K. R. Goodearl, Introduction to the Second Edition, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, Template:W, 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 $I_1\subseteq \cdots \subseteq I_{k-1}\subseteq I_k\subseteq I_{k+1}\subseteq \cdots$ is such a chain, then there exists an n such that $I_n=I_{n+1}=\cdots .$
• 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 Template:M, which deals with commutative rings and related structures, and the more general Template:M, in which commutativity is not assumed in the structures studied (i.e., the theory potentially applies to both commutative and noncommutative structures).

Hyponyms

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Derived terms

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Related terms

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Further reading

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