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...

finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of...

property. For Noetherian local rings, there is the following chain of inclusions. Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete...

length. Noetherian objects are named after Emmy Noether, who was the first to study the ascending and descending chain conditions for rings. Specifically:...

ideals have this property are called Noetherian rings. Every field, and the ring of integers are Noetherian rings. So, the theorem can be generalized and...

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...

commutative ring R is Noetherian, the same is true for every polynomial ring over it, and for every quotient ring, localization, or completion of the ring. The...

Artinian ring is Artinian. Since an Artinian ring is also a Noetherian ring, and finitely-generated modules over a Noetherian ring are Noetherian, it is...

full matrix ring M n ( R ) {\displaystyle M_{n}(R)} over a left Artinian (resp. left Noetherian) ring R is left Artinian (resp. left Noetherian). The following...

quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called an excellent ring if it is...

commutative rings, similar to the role of the finite-dimensional vector spaces in linear algebra. In particular, Noetherian rings (see also § Noetherian rings, below)...

extensions, and turn out to be minimal injective extensions. Over a Noetherian ring, every injective module is uniquely a direct sum of indecomposable...

ring Local ring Noetherian and artinian rings Ordered ring Poisson ring Reduced ring Regular ring Ring of periods SBI ring Valuation ring and discrete...

of the ring; by the Krull intersection theorem, this is the case for any commutative Noetherian ring which is an integral domain or a local ring. There...

Noetherian module. Any module that is finite as a set is Noetherian. Any finitely generated right module over a right Noetherian ring is a Noetherian...

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...

In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring R with finite injective dimension as an R-module. There are many...

required that a local ring be (left and right) Noetherian, and (possibly non-Noetherian) local rings were called quasi-local rings. In this article this...

commutative Noetherian ring, then Spec(R), the prime spectrum of R, is a Noetherian topological space. More generally, a Noetherian scheme is a Noetherian topological...

modules and coherent modules all of which are defined below. Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent...

noncommutative rings, especially noncommutative Noetherian rings. For the definitions of a ring and basic concepts and their properties, see Ring (mathematics)...

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...

In commutative algebra, a G-ring or Grothendieck ring is a Noetherian ring such that the map of any of its local rings to the completion is regular (defined...

In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal...

Goldie's theorem applies to semiprime right Noetherian rings, since by definition right Noetherian rings have the ascending chain condition on all right...

example, finite direct sums of Noetherian rings are Noetherian, as is the ring of formal power series over a Noetherian ring. Another application of such...

in a Noetherian ring R {\displaystyle R} is called a perfect ideal if its grade equals the projective dimension of the associated quotient ring, ⁠ grade...

of the ring. In general, if the nilradical is finitely generated (e.g., the ring is Noetherian), then it is nilpotent. For noncommutative rings, there...

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...

catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings Suppose that A is a Noetherian domain and...