finitely generated module

A module X over a ring R is said to be finitely generatedMathworldPlanetmathPlanetmath 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 combinationsMathworldPlanetmath of elements of S; moreover, let us recall that the span of the empty setMathworldPlanetmath is defined to be the singleton consisting of only one vector, the zero vectorMathworldPlanetmath 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. 1.

    Rx={rxrR} is a cyclic R-module generated by {x}.

  2. 2.

    RRx is a finitely-generated R-module generated by {1,x}. Any element in RRx can be expressed uniquely as r+sx.

  3. 3.

    R[x] is not finitely generated as an R-module. For if there is a finite setMathworldPlanetmath Y R[x], taking d to be the largest of all degrees of polynomials in Y, then xd+1 would not be in the of Y, assumed to be R[x], which is a contradictionMathworldPlanetmathPlanetmath. (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