A module 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.