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...

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...

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:...

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...

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

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

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...

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...

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...

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

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

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...

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

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...

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...

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...

and his former advisor Emil Artin, states: Let A be a commutative Noetherian ring and B ⊂ C {\displaystyle B\subset C} commutative algebras over A. If...

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...

the Lasker–Noether primary decomposition of ideals in commutative Noetherian rings. Specifically, if an ideal J is decomposed as a finite intersection...

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

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

interpolates between R and its associated graded ring grIR. Assume R is Noetherian; then R[It] is also Noetherian. The Krull dimension of the Rees algebra is...

Rings are assumed to be commutative except in the last section on dimensions of non-commutative rings. Let R be a noetherian ring or valuation ring....

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...

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...