modules are a generalization of vector spaces

A http://planetmath.org/node/1022module is the natural generalization of a vector space, 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 $f:M\to N$ is called an $R$-morphism (or homomorphism) if:

$$\forall x,y\in M:f(x+y)=f(x)+f(y)\mathit{\hspace{1em}}\mathrm{and}\mathit{\hspace{1em}}\forall x\in M\forall \lambda \in R:f(\lambda x)=\lambda f(x)$$

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 $\mathrm{Im}f:=\{f(x):x\in M\}$ and kernel $\mathrm{Ker}f:=\{x\in M:f(x)=0_N\}$ 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