Suppose is an -module morphism. Let be the usual kernel of when it is viewed as simply a group morphism, i.e.., . This set is not only a subgroup of (when viewed as an abelian group), but also a submodule of . It is still called the kernel of the morphism . One can also give a definition of the kernel without reference to any elements, using the zero morphism.
Images are submodules
Similarly, let be the usual image of (as a set map or group morphism). As with the kernel, this set is not just a subgroup of (when viewed as abelian group), but also a submodule of . It is still called the image of the morphism .
Hom-sets? More like hom-modules!
For each pair of -modules and , we can consider the set of all -module morphisms from to . This set has a natural (!) structure of an abelian group; when is commutative, it has the structure of an -module.
When , the set has the natural structure of a ring with unity. It is called the endomorphism ring of and is sometimes denoted ; when is commutative, the ring has the natural structure of an -algebra.