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Concept
Noetherian
Summary
In mathematics, the adjective Noetherian is used to describe that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite length. Noetherian objects are named after Emmy Noether, who was the first to study the ascending and descending chain conditions for rings. Specifically:
Noetherian group, a group that satisfies the ascending chain condition on subgroups.
Noetherian ring, a ring that satisfies the ascending chain condition on ideals.
Noetherian module, a module that satisfies the ascending chain condition on submodules.
More generally, an object in a is said to be Noetherian if there is no infinitely increasing filtration of it by subobjects. A category is Noetherian if every object in it is Noetherian.
Noetherian relation, a binary relation that satisfies the ascending chain condition on its elements.
Noetherian topological space, a topological space that satisfies the descending chain condition on closed sets.
Noetherian induction, also called well-founded induction, a proof method for binary relations that satisfy the descending chain condition.
Noetherian rewriting system, an abstract rewriting system that has no infinite chains.
Noetherian scheme, a scheme in algebraic geometry that admits a finite covering by open spectra of Noetherian rings.
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Related concepts (7)
Noetherian
In mathematics, the adjective Noetherian is used to describe that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite length. Noetherian objects are named after Emmy Noether, who was the first to study the ascending and descending chain conditions for rings. Specifically: Noetherian group, a group that satisfies the ascending chain condition on subgroups.
Noetherian module
In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion. Historically, Hilbert was the first mathematician to work with the properties of finitely generated submodules. He proved an important theorem known as Hilbert's basis theorem which says that any ideal in the multivariate polynomial ring of an arbitrary field is finitely generated.
Abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning.
Related courses (1)
MATH-215: Rings and fields
C'est un cours introductoire dans la théorie d'anneau et de corps.
Related lectures (10)
Local Rings and Minimal Primes
Explores local rings, Noetherian rings, and minimal primes in the context of integral domains.
Irreducible Factors and Noetherian Rings
Discusses irreducible factors in rings and the properties of Noetherian rings.
Discrete Valuation Rings
Explores discrete valuation rings, their properties, uniqueness of representation, and relationship with principal ideal domains.