# 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:

 $\forall x,y\in M:f(x+y)=f(x)+f(y)\quad\mathrm{and}\quad\forall x\in M\forall% \lambda\in R:f(\lambda x)=\lambda f(x)$

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 ModulesAreAGeneralizationOfVectorSpaces 2013-03-22 13:38:18 2013-03-22 13:38:18 jgade (861) jgade (861) 7 jgade (861) Example msc 15A99