Suppose $f:M\to N$ is an $R$-module morphism. Let $\mathrm{ker}(f)$ be the usual kernel of $f$ when it is viewed as simply a group morphism, i.e.., $\mathrm{ker}(f)=\{m\in M\mid f(m)={0}_{N}\}$. This set is not only a subgroup of $M$ (when viewed as an abelian group), but also a submodule of $M$. It is still called the kernel of the morphism $f$. One can also give a definition of the kernel without reference to any elements, using the zero morphism.

Images are submodules

Similarly, let $\mathrm{im}(f)$ be the usual image of $f$ (as a set map or group morphism). As with the kernel, this set is not just a subgroup of $N$ (when viewed as abelian group), but also a submodule of $N$. It is still called the image of the morphism $f$.

Hom-sets? More like hom-modules!

For each pair of $R$-modules $M$ and $N$, we can consider the set ${\mathrm{Hom}}_{R}(M,N)$ of all $R$-module morphisms from $M$ to $N$. This set has a natural (!) structure of an abelian group; when $R$ is commutative, it has the structure of an $R$-module.

When $N=M$, the set ${\mathrm{Hom}}_{R}(M,M)$ has the natural structure of a ring with unity. It is called the endomorphism ring of $M$ and is sometimes denoted ${\mathrm{End}}_{R}(M)$; when $R$ is commutative, the ring ${\mathrm{End}}_{R}(M)$ has the natural structure of an $R$-algebra.