Noetherian
Encyclopedia
In mathematics, the adjective
Noetherian is used to describe objects that satisfy an ascending or descending chain condition
on certain kinds of subobjects; in particular,
See also:
Adjective
In grammar, an adjective is a 'describing' word; the main syntactic role of which is to qualify a noun or noun phrase, giving more information about the object signified....
Noetherian is used to describe objects that satisfy an ascending or descending chain condition
Ascending chain condition
The ascending chain condition and descending chain condition are finiteness properties satisfied by some algebraic structures, most importantly, ideals in certain commutative rings...
on certain kinds of subobjects; in particular,
- Noetherian group, a group that satisfies the ascending chain condition on subgroups
- Noetherian ringNoetherian ringIn mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...
, a ring that satisfies the ascending chain condition on ideals. - Noetherian moduleNoetherian moduleIn abstract algebra, an Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion....
, a module that satisfies the ascending chain condition on submodules. - Noetherian topological spaceNoetherian topological spaceIn mathematics, a Noetherian topological space is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they are the complements of the closed subsets...
, a topological space that satisfies the descending chain condition on closed sets. - Noetherian induction, also called well-founded induction.
- Noetherian rewriting system, an abstract rewriting systemAbstract rewriting systemIn mathematical logic and theoretical computer science, an abstract rewriting system is a formalism that captures the quintessential notion and properties of rewriting systems...
that has no infinite chains - Noetherian scheme.
See also:
- Emmy NoetherEmmy NoetherAmalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of...
, who was the first to study the ascending and descending chain conditions for rings, and for whom the term is named. - Artinian ringArtinian ringIn abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
, a ring that satisfies the descending chain condition on ideals.