Noetherian ring

plural Noetherian rings · alternative noetherian ring

1. ring theoryalgebraA 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).

   The central position occupied by Noetherian rings in commutative ring theory became evident from her^([Noether's]) work.

   1986, Hideyuki Matsumura, translated by M. Reid, Commutative Ring Theory, Paperback edition, Cambridge University Press, published 1989, page ix

   2 more examples

   In this chapter the focus moves from semiprime rings to general Noetherian rings, although it does concentrate on prime and semiprime ideals.

   2000, John C. McConnell, James Christopher Robson, Lance W. Small, Noncommutative Noetherian Rings, 2nd edition, American Mathematical Society, page 97

   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.