Noetherian ring

English Alternative forms en noetherian ring Etymology Named after German mathematician Emmy Noether (1882–1935). Pronunciation en /nəˈtɛ.ɹi.ən ˈɹɪŋɡ/ Noun en 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). en 1986 M. Reid Hideyuki Matsumura Commutative Ring Theory Paperback https://books.google.com.au/books?id=yJwNrABugDEC&pg=PR9&dq=%22Noetherian+ring%22%7C%22Noetherian+rings%22&hl=en&sa=X&ved=0ahUKEwjn386_ks7dAhWK7Z8KHXvlBYwQ6AEI5QIwOQ#v=onepage&q=%22Noetherian%20ring%22%7C%22Noetherian%20rings%22&f=false ix w:Cambridge University Press 1989 The central position occupied by Noetherian rings in commutative ring theory became evident from her[Noether's] work. en 2000 John C. McConnell; James Christopher Robson; Lance W. Small Noncommutative Noetherian Rings 2nd w:American Mathematical Society https://books.google.com.au/books?id=DcTpBwAAQBAJ&printsec=frontcover&dq=%22Noetherian+ring%22%7C%22Noetherian+rings%22&hl=en&sa=X&ved=0ahUKEwjn386_ks7dAhWK7Z8KHXvlBYwQ6AEINTAC#v=onepage&q=%22Noetherian%20ring%22%7C%22Noetherian%20rings%22&f=false 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 is such a chain, then there exists an n such that 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 en commutative algebra, which deals with commutative rings and related structures, and the more general en noncommutative algebra, in which commutativity is not assumed in the structures studied (i.e., the theory potentially applies to both commutative and noncommutative structures). Hyponyms en Noetherian domain en Dedekind domain en Artinian ring Derived terms en left Noetherian ring en right Noetherian ring Related terms en left-Noetherian en Noetherian en right-Noetherian Further reading Ascending chain condition on principal ideals Ascending chain condition Artinian ring Dedekind domain en eponyms