finitely generated module
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 $X$. Let us recall that the span of a (not necessarily finite) set $X$ of vectors is the class of all (finite) linear combinations^{} of elements of $S$; moreover, let us recall that the span of the empty set^{} is defined to be the singleton consisting of only one vector, the zero vector^{} $\overrightarrow{0}$. A module $X$ is then called cyclic if it can be a singleton.
Examples. Let $R$ be a commutative ring with 1 and $x$ be an indeterminate.

1.
$Rx=\{rx\mid r\in R\}$ is a cyclic $R$module generated by $\{x\}$.

2.
$R\oplus Rx$ is a finitelygenerated $R$module generated by $\{1,x\}$. 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^{} $Y$ $R[x]$, taking $d$ to be the largest of all degrees of polynomials in $Y$, then ${x}^{d+1}$ would not be in the of $Y$, assumed to be $R[x]$, which is a contradiction^{}. (Note, however, that $R[x]$ is finitelygenerated as an $R$algebra.)
Title  finitely generated module 

Canonical name  FinitelyGeneratedModule 
Date of creation  20130322 14:01:08 
Last modified on  20130322 14:01:08 
Owner  Thomas Heye (1234) 
Last modified by  Thomas Heye (1234) 
Numerical id  15 
Author  Thomas Heye (1234) 
Entry type  Definition 
Classification  msc 16D10 
Related topic  ModuleFinite 
Related topic  Span 
Defines  finitely generated 
Defines  cyclic module 