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 and are -modules then a mapping is called an -morphism (or homomorphism) if:
Note as mentioned in the beginning, if is a field, these properties are the defining properties for a linear transformation.
Similarly in vector space terminology the image and kernel are called the range and null-space respectively.