Intuition for morphisms of modules Let $A$ be a ring. We introduce the notation$$\text{Hom}(M, N) = \{A\text{-module morphisms }M \to N\}.$$Note that $\text{Hom}_A(M, N)$ is an abelian group (under pointwise addition of functions). If $A$ is commutative, then $\text{Hom}_A(M, N)$ has a natural $A$-module structure: for $c \in A$ and $f \in \text{Hom}_A(M, N)$, we define $cf \in \text{Hom}_A(M, N)$ by $(cf)(m) = c \cdot f(m)$. When $A$ is not commutative, then the requirement that module morphisms satisfy $g(am) = a \cdot g(m)$, $\forall a \in A$, may fail for $cf$, in general, unless $c$ is in the center of $A$.
For any module $M$ and an element $m \in M$, the assignment $f_m : x \mapsto xm$ gives a morphism $f_m: A \to M$, of $A$-modules. This yields a natural isomorphism$$\text{Hom}_A(A, M) \leftrightarrow M,\text{ }f \mapsto f(1_A),\text{ }m \mapsto [f_m: x \mapsto xm]$$of abelian groups. Note that, in general, the above maps are not morphisms of modules, ulnless $A$ is commutative since $\text{Hom}_A(A, M)$ does't have the structure of a left $A$-module in general.
Now, in spite all of that, I still feel like I do not have very good intuition for morphisms of modules. Could anyone lend me their intuition for thinking about/working with them?
EDIT: I am mainly interested in the intuition for such objects in context of representation theory and noncommutative algebra.
 A: I only have the algebro-geometric "commutative" intuition, and from that point of view modules are generalizations of vector bundles over $Spec A$, and flat modules are actual vector bundles --- localisation of $M$ at a prime ideal of $A$ is free, and this is a sufficient condition for flatness too (for rigour, one probably has to throw in some Noetherianity/finiteness assumptions).
Now for vector bundles there is a covariant functor that sends a vector bundle to its module of sections (by the way you can think of forming the module $Hom_A(A,M)$ as taking sections). Unfortunately, one cannot extend this functor to general modules.
I have encountered a description of another duality in an old book of Fischer on Complex Analytic Geometry. Suppose we work over a field now, so $A$ is a $k$-algebra, and suppose $Y$ is a scheme over $X=Spec A$ together with two operations (over $X$) $+: Y \times_X Y \to Y$ and $\cdot: \mathbb{A}^1_k \times Y \to Y$, such that on $k$-points, the fibres of $Y$ over $X$ are $k$-vector spaces, with addition of vectors and multiplication by scalars defined by $+$ and $\cdot$. Call spaces like $Y$ linear spaces, and define morphisms between them as maps that respect the two operations. Then there is a contravariant functor from linear spaces to modules: send a linear space $Y$ to $Hom_{LS}(Y, \mathbb{A^1}\times X)$, the latter has a structure of $A$-module via operations $+$ and $\cdot$; one can see that this functor is an equivalence of categories.
