Noetherian ring

Etymology
Named after German mathematician (1882–1935).

Noun

 * 1)  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).
 * 2) * 2004, K. R. Goodearl, Introduction to the Second Edition, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings,, 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.
 * 1) * 2004, K. R. Goodearl, Introduction to the Second Edition, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings,, 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, which deals with commutative rings and related structures, and the more general , in which commutativity is not assumed in the structures studied (i.e., the theory potentially applies to both commutative and noncommutative structures).