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 finitely-generated $R$-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 finitely-generated as an $R$-algebra.)

Title	finitely generated module
Canonical name	FinitelyGeneratedModule
Date of creation	2013-03-22 14:01:08
Last modified on	2013-03-22 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