Noetherian module

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.

See also

• Artinian module
• Ascending/Descending chain condition
• Composition series
• finitely generated module
• Krull dimension

References

• Eisenbud Commutative Algebra with a View Toward Algebraic Geometry, Springer-Verlag, 1995.

This algebra-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Module theory Commutative algebra Algebra stubs Wikimedia Foundation. 2010. Игры ⚽ Поможем сделать НИР Alfred Knopf Monahans Look at other dictionaries: 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. Equivalently, a ring is Noetherian if it… …   Wikipedia Noetherian — 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, Noetherian group, a group that satisfies the ascending chain condition on… …   Wikipedia Module (mathematics) — For other uses, see Module (disambiguation). 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. Modules also… …   Wikipedia noetherian — adjective a) Of a ring in which any ascending chain of ideals eventually starts repeating. b) Of a module in which any ascending chain of submodules eventually starts repeating …   Wiktionary Finitely-generated module — In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated R module also may be called a finite R module or finite over R.[1] Related concepts include finitely cogenerated modules, finitely… …   Wikipedia Artinian module — In abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its 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… …   Wikipedia Radical of a module — In mathematics, in the theory of modules, the radical of a module is a component in the theory of structure and classification. It is a generalization of the Jacobson radical for rings. In many ways, it is the dual notion to that of the socle… …   Wikipedia Injective module — In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z module Q of all rational numbers. Specifically, if Q is a submodule of some… …   Wikipedia Serial module — Chain ring redirects here. For the bicycle part, see Chainring. In abstract algebra, a uniserial module M is a module over a ring R, whose submodules are totally ordered by inclusion. This means simply that for any two submodules N1 and N2 of M,… …   Wikipedia Finitely generated module — In mathematics, a finitely generated module is a module that has a finite generating set. Equivalently, it is a homomorphic image of a free module on finitely many generators. The kernel of this homomorphism need not be finitely generated (then… …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”