Noetherian ring


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Noetherian ring

[‚nō·ə′thir·ē·ən′riŋ]
(mathematics)
A ring is Noetherian on left ideals (or right ideals) if every ascending sequence of left ideals (or right ideals) has only a finite number of distinct members.
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He focuses on deformations over local complete Noetherian rings, which covers most types of deformations of algebraic structures that working mathematicians meet in their professional life.
Coverage includes a guide to closure operations in commutative algebra, a survey of test ideals, finite-dimensional vector spaces with Frobenius action, finiteness and homological conditions in commutative group rings, regular pullbacks, noetherian rings without finite normalization, Krull dimension of polynomial and power series rings, the projective line over the integers, on zero divisor graphs, and a closer look at non-unique factorization via atomic decay and strong atoms.
In his treatment of affine algebras and Noetherian rings he describes the Galois theory of fields, algebras and affine fields, transcendence degree and the Krull dimension of a ring, modules and rings satisfying chain conditions, localization in the prime spectrum, the Krull dimension theory of commutative Noetherian rings.
Hereditary Noetherian prime rings may be the only non-commutative Noetherian rings whose projective modules, both finitely and infinitely generated, have nontrivial direct sum behavior and a structure theorem describing that behavior, say mathematicians Levy (U.
of Iowa), others have added to that work through the introduction of tilting modules and the tilting theorem for finitely generated modules over artin algebras, torsion theories in the categories of modules over two rings and a pair of equivalences between the torsion and torsion-free parts of the torsion theories, a generalization of Morita duality, notions of cotilting modules and a cotilting theorem for noetherian rings.
An introduction to noncommutative noetherian rings, 2d ed.