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In mathematics, more specifically in the area of abstract algebra known as ring theory, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals; that is, given any chain of ideals: : there exists an ''n'' such that: : There are other equivalent formulations of the definition of a Noetherian ring and these are outlined later in the article. Noetherian rings are named after Emmy Noether. 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. For instance, the ring of integers and the polynomial ring over a field are both Noetherian rings, and consequently, such theorems as the Lasker–Noether theorem, the Krull intersection theorem, and the Hilbert's basis theorem hold for them. Furthermore, if a ring is Noetherian, then it satisfies the descending chain condition on ''prime ideals''. This property suggests a deep theory of dimension for Noetherian rings beginning with the notion of the Krull dimension. == Characterizations == For noncommutative rings, it is necessary to distinguish between three very similar concepts: * A ring is left-Noetherian if it satisfies the ascending chain condition on left ideals. * A ring is right-Noetherian if it satisfies the ascending chain condition on right ideals. * A ring is Noetherian if it is both left- and right-Noetherian. For commutative rings, all three concepts coincide, but in general they are different. There are rings that are left-Noetherian and not right-Noetherian, and vice versa. There are other, equivalent, definitions for a ring ''R'' to be left-Noetherian: * Every left ideal ''I'' in ''R'' is finitely generated, i.e. there exist elements ''a''1, ..., ''an'' in ''I'' such that ''I'' = ''Ra''1 + ... + ''Ra''''n''.〔Lam (2001), p. 19〕 * Every non-empty set of left ideals of ''R'', partially ordered by inclusion, has a maximal element with respect to set inclusion.〔 Similar results hold for right-Noetherian rings. For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated. (The result is due to I. S. Cohen.) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Noetherian ring」の詳細全文を読む スポンサード リンク
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