modules are a generalization of vector spaces

A is the natural generalizationPlanetmathPlanetmath of a vector spaceMathworldPlanetmath, in fact, when working over a field it is just another word for a vector space.

If M and N are R-modules then a mapping f:MN is called an R-morphism (or homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath) if:


Note as mentioned in the beginning, if R is a field, these properties are the defining properties for a linear transformation.

Similarly in vector space terminology the image Imf:={f(x):xM} and kernel Kerf:={xM:f(x)=0N} are called the range and null-space respectively.

Title modules are a generalization of vector spaces
Canonical name ModulesAreAGeneralizationOfVectorSpaces
Date of creation 2013-03-22 13:38:18
Last modified on 2013-03-22 13:38:18
Owner jgade (861)
Last modified by jgade (861)
Numerical id 7
Author jgade (861)
Entry type Example
Classification msc 15A99