In mathematics, the adjective Noetherian is used to describe objects 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 category 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.
In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects...
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied...
is a Noetherian ring. More generally, a scheme is locally Noetherian if it is covered by spectra of Noetherian rings. Thus, a scheme is Noetherian if and...
In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially...
In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition...
Noetherian and Artinian. Homomorphic images and subgroups of Noetherian groups are Noetherian, and an extension of a Noetherian group by a Noetherian...
rings over a field are Noetherian is called Hilbert's basis theorem. Moreover, many ring constructions preserve the Noetherian property. In particular...
deduce theorems of interest for usual schemes. A locally Noetherian scheme is a locally Noetherian formal scheme in the canonical way: the formal completion...
discrete valuation ring. A noetherian ring is a Krull domain if and only if it is an integrally closed domain. In the non-noetherian setting, one has the following:...
Since an Artinian ring is also a Noetherian ring, and finitely-generated modules over a Noetherian ring are Noetherian, it is true that for an Artinian...
chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over...
of a Noetherian object. For example, finite direct sums of Noetherian rings are Noetherian, as is the ring of formal power series over a Noetherian ring...
as a Noetherian chain, and a function constructed as a polynomial in this chain is called a Noetherian function. So, for example, a Noetherian chain...
In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called...
In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection...
finite-dimensional vector spaces in linear algebra. In particular, Noetherian rings (see also § Noetherian rings, below) can be defined as the rings such that every...
theorem, introduced by Cohen (1946), describes the structure of complete Noetherian local rings. Some consequences of Cohen's structure theorem include three...
over a Noetherian ring R is Noetherian. Both facts imply that a finitely generated commutative algebra over a Noetherian ring is again a Noetherian ring...
are all zero-divisors is a Noetherian ring.: 53 Every overring of a Krull 1-dimensional Noetherian domain is a Noetherian ring.: 53 These statements...
theory of Noetherian rings. By a theorem of Jean-Pierre Serre, global dimension can be used to characterize within the class of commutative Noetherian local...
a left (resp. right) Noetherian ring. This is not true for general modules; that is, an Artinian module need not be a Noetherian module. An integral domain...
authors required that a local ring be (left and right) Noetherian, and (possibly non-Noetherian) local rings were called quasi-local rings. In this article...
better understanding of noncommutative rings, especially noncommutative Noetherian rings. For the definitions of a ring and basic concepts and their properties...
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