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Concept# Artinian module

Summary

In mathematics, specifically 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 (with left or right multiplication). Both concepts are named for Emil Artin.
In the presence of the axiom of (dependent) choice, the descending chain condition becomes equivalent to the minimum condition, and so that may be used in the definition instead.
Like Noetherian modules, Artinian modules enjoy the following heredity property:

- If M is an Artinian R-module, then so is any submodule and any quotient of M. The converse also holds:
- If M is any R-module and N any Artinian submodule such that M/N is Artinian, then M is Artinian. As a consequence, any finitely-generated module over an Artinian ring is Artinian. Since an Artinian ring is also a Noetherian ring, and finitely-generated modules over a N

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MATH-311: Rings and modules

The students are going to solidify their knowledge of ring and module theory with a major emphasis on commutative algebra and a minor emphasis on homological algebra.

We define and study a lift of the Boardman-Vogt tensor product of operads to bimodules over operad

2014Related lectures (1)

In this PhD thesis, we construct an explicit algebraic model over Z of the cochains of the free loop space of a 1-connected space X. We start from an enriched Adams-Hilton model of X, which can be obtained relatively easily when X is the realisation of a simplicial set. Note it is not supposed that the Steenrod algebra acts trivially on X. The second part is dedicated to the construction of a model of the cochains of mapping spaces XY. where X is r-connected and Y is a CW-complex that has dimension less or equal to r. The space X must possess commutative models for the cochains of each Ωk X for k ≤ r. We first construct an algebraic model for the cochains of XSn ∀n ≤ r, then we then glue all of them to obtain a model of the cochains of XY. We give examples for each of these situations. The techniques used here rely heavily on the concept of a twisted bimodule. A description of this can be found in [DH99b].

Related concepts (7)

Finitely generated module

In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring R may also be called a finite R-module, finite over R, or a module of

Semisimple module

In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts.

Commutative ring

In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebr