I've been trying for a while to get a real concrete handle on the relationship between representations and modules. To frame the question, I'll put here the standard situation I have in mind:


A ring $R$ lives in the category Ab of Abelian groups as an internal monoid $(\mu\_R, \eta\_R)$. A module is then just an Abelian group $A$ and a map $m : R \otimes A \rightarrow A$ that commutes with the monoid structure in the way you'd expect.


Alternatively, take an Abelian group $A$ and look at its group of endomorphisms $[A,A]$. This has an internal monoid $(\mu\_A, \eta\_A)$ just taking composition and identity. Then a representation is just a monoid homomorphism $(R, \mu\_R, \eta\_R) \rightarrow (A, \mu\_A, \eta\_A)$ in Ab. I.e. a ring homomorphism.


But then, Ab is monoidal closed, so these are the same concept under the iso


$$\hom(R\otimes A, A) \cong \hom(R, [A,A])$$


This idea seems to work for any closed category where one wants to relate a multiplication to composition. So, my question is, since these things are isomorphic in such a general context, why are they taught as two separate concepts? Is it merely pedagogical, or are there useful examples where modules and representations are distinct?



When are modules and representations not the same thing?
Here is my representation theorist's perspective: the key difference between representations and modules is that representations are "non-linear", whereas modules are "linear". I'll concentrate on the case of groups as the most familiar, but this applies more generally. 


As Greg has already mentioned, in the most general sense, a representation is a homomorphism $f:G\to H,$ and usually there is no linear (or additive) structure on $H$, i.e. the set $f(g)$ need not be closed under sums; in fact, if $H$ is a non-abelian group, e.g. the symmetric group, the notion of sum doesn't even make sense (if $H=GL(V)$ then we may view its elements as endomorphisms of $V$ and add them, but this is unnatural since, by definition, $f$ is compatible with *multiplicative* structure). By contrast, a module involves a *linear* action $G\times V\to V,$ which is then "completed" by allowing arbitrary linear combinations, leading to certain technical advantages.


Here is an example of a construction that is very useful and makes perfect sense module-theoretically, but not representation-theoretically: change of scalars. Given a module $M$ over a group ring $R[G]$ and a commutative ring homomorphism $R\to S,$ one gets a module $S\otimes\_R M$ over the group ring $S[G]$. Common examples involve extensions of scalars (e.g. from $\mathbb{R}$ to $\mathbb{C}$, from a field $K$ of definition to the splitting field, from $\mathbb{Z}$ to $\mathbb{Z}\_p$) and, more to the point, reductions (e.g. from $\mathbb{Z}$ or $\mathbb{Z}\_p$ to $\mathbb{Z}/p\mathbb{Z}$). The module language is, predictably, also very useful in providing categorical descriptions of various operations on representations, such as functors of induction and restriction, 


$$Ind\_H^G: H\text{-mod}\to G\text{-mod}\ \text{ and }\ Res\_H^G: G\text{-mod}\to H\text{-mod},$$ 


where $H$ is a subgroup of $G,$ or the monoidal structure on $G$-mod.


Finally, here are two illustrations of the complementary nature of the two approaches besides the group case, in linear algebra. A single linear transformation $T:V\to V$ on a finite-dimensional vector space $V$ over $K$ is most naturally viewed as a representation (no additive structure); in this case, it's a representation of the quiver with a single vertex and a single loop. From this point of view, classification up to isomorphism is a problem about conjugacy classes of linear transformations, 


$$T\to gTg^{-1},\ g\in GL(V).$$ 


By contrast, in the module style description we associate with $T$ a module over the ring $K[x]$ of polynomials in one variable over $K$ and classification problem reduces to the structure of modules over $K[x]$, which is a PID, with all the usual consequences. (*Here the module picture is more illuminating.*) If we consider a linear operator $S:V\to W$ between two different vector spaces, 


$$S\to hSg^{-1},\ g\in GL(V),\ h\in GL(W),$$


and a classification up to isomorphism is accomplished by row and column reduction. The corresponding quiver $\circ\to\circ$ is a single arrow connecting two distinct vertices, but its path algebra is less familiar. (*Here the representation theory picture is more illuminating.*)


