PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (204)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Low Entry average rating: No information on entry rating
finitely generated module (Definition)

A module $ X$ over a ring $ R$ is said to be finitely generated if there is a finite subset $ Y$ of $ X$ such that $ Y$ spans $ X$. A module $ X$ is cyclic if it can be spanned by a singleton.

Examples. Let $ R$ be a commutative ring with 1 and $ x$ be an indeterminate.

  1. $ Rx=\lbrace rx \mid r\in R \rbrace$ is a cyclic $ R$-module generated by $ \lbrace x \rbrace$.
  2. $ R\oplus Rx$ is a finitely-generated $ R$-module generated by $ \lbrace 1, x \rbrace$. Any element in $ R\oplus Rx$ can be expressed uniquely as $ r+sx$.
  3. $ R[x]$ is not finitely generated as an $ R$-module. For if there is a finite set $ Y$ spanning $ R[x]$, taking $ d$ to be the largest of all degrees of polynomials in $ Y$, then $ x^{d+1}$ would not be in the spanning set of $ Y$, assumed to be $ R[x]$, which is a contradiction. (Note, however, that $ R[x]$ is finitely-generated as an $ R$-algebra.)



Anyone with an account can edit this entry. Please help improve it!

"finitely generated module" is owned by Thomas Heye. [ full author list (5) | owner history (1) ]
(view preamble)

View style:

See Also: module-finite

Also defines:  finitely generated, cyclic module
Log in to rate this entry.
(view current ratings)

Cross-references: contradiction, polynomials, degrees, finite set, generated by, indeterminate, commutative ring, singleton, spans, subset, finite, ring, module
There are 32 references to this entry.

This is version 10 of finitely generated module, born on 2003-10-15, modified 2007-06-29.
Object id is 4957, canonical name is FinitelyGeneratedRModule.
Accessed 5495 times total.

Classification:
AMS MSC: 16D10 (Associative rings and algebras :: Modules, bimodules and ideals :: General module theory)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)