Noetherian module
Encyclopedia
In 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.
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. However, the property is named after Emmy Noether
who was the first one to discover the true importance of the property.
Two other equivalent conditions are: a module is Noetherian if and only if all of its submodules are finitely generated, if and only if any nonempty set S of submodules has a maximal element (by inclusion).
A right Noetherian ring
R is, by definition, a Noetherian right R module over itself using multiplication on the right. Likewise a ring is called left Noetherian ring when R is Noetherian considered as a left R module. When R is a commutative ring
the left-right adjectives may be dropped, as they are unnecessary. Also, if R is Noetherian on both sides, it is customary to call it Noetherian and not "left and right Noetherian".
Any finitely generated right module over a right Noetherian ring is a Noetherian module.
If M is a module and K a submodule, then M is Noetherian if and only if K and M/K are Noetherian. This is in contrast to the general situation with finitely generated modules: a submodule of a finitely generated module need not be finitely generated.
The Noetherian condition can also be defined on bimodule structures as well: a Noetherian bimodule is a bimodule whose poset of sub-bimodules satisfies the ascending chain condition. Since a sub-bimodule of an R-S bimodule M is a fortiori a left R-module, if M considered as a left R module were Noetherian, then M is automatically a Noetherian bimodule. It may happen, however, that a bimodule is Noetherian without its left or right structures being Noetherian.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, an Noetherian module is a module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
that satisfies the ascending 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 its submodules, where the submodules are partially ordered by inclusion.
Historically, Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
was the first mathematician to work with the properties of finitely generated submodules. He proved an important theorem known as Hilbert's basis theorem
Hilbert's basis theorem
In mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated. This can be translated into algebraic geometry as follows: every algebraic set over a field can be described as the...
which says that any ideal in the multivariate polynomial ring of an arbitrary field is finitely generated. However, the property is named after Emmy Noether
Emmy Noether
Amalie 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 one to discover the true importance of the property.
Two other equivalent conditions are: a module is Noetherian if and only if all of its submodules are finitely generated, if and only if any nonempty set S of submodules has a maximal element (by inclusion).
A right Noetherian ring
Noetherian ring
In 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...
R is, by definition, a Noetherian right R module over itself using multiplication on the right. Likewise a ring is called left Noetherian ring when R is Noetherian considered as a left R module. When R is a commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
the left-right adjectives may be dropped, as they are unnecessary. Also, if R is Noetherian on both sides, it is customary to call it Noetherian and not "left and right Noetherian".
Any finitely generated right module over a right Noetherian ring is a Noetherian module.
If M is a module and K a submodule, then M is Noetherian if and only if K and M/K are Noetherian. This is in contrast to the general situation with finitely generated modules: a submodule of a finitely generated module need not be finitely generated.
The Noetherian condition can also be defined on bimodule structures as well: a Noetherian bimodule is a bimodule whose poset of sub-bimodules satisfies the ascending chain condition. Since a sub-bimodule of an R-S bimodule M is a fortiori a left R-module, if M considered as a left R module were Noetherian, then M is automatically a Noetherian bimodule. It may happen, however, that a bimodule is Noetherian without its left or right structures being Noetherian.
See also
- Artinian moduleArtinian moduleIn abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings are for rings, and a ring is Artinian if and only if it is an Artinian module over itself...
- Ascending/Descending chain conditionAscending chain conditionThe ascending chain condition and descending chain condition are finiteness properties satisfied by some algebraic structures, most importantly, ideals in certain commutative rings...
- Composition seriesComposition seriesIn abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are not semisimple, hence...
- finitely generated module
- Krull dimensionKrull dimensionIn commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....