are representations $V$ “satisfying the relations in $I$”, that is, such that for any linear combination of paths $\sum_i \lambda_iw_i$ lying in $I$, we have that $\sum_i \lambda_i V_{w_i} = 0$.

One of the main motivations for studying representations of quivers can be summarized in the following results. First, representations of a quiver and modules over its path algebra should be viewed as being the same thing. More precisely:

\[prop::Morita\] Let $Q$ be a quiver and $I$ be a two-sided ideal of $kQ$. Then the categories ${\mbox{{\rm Mod \!}}}(kQ/I)$ and ${\mbox{{\rm Rep \!\!}}}(Q^{op},I^{op})$ are equivalent. (Here ${\mbox{{\rm Mod \!}}}A$ is the category of right(!) modules over $A$, and $Q^{op}$ is the opposite quiver, obtained by reversing the orientation of all arrows of $Q$).

The same is true of ${\mbox{{\rm mod \!}}}(kQ/I)$ and ${\mbox{{\rm rep \!\!}}}(Q^{op},I^{op})$, the full subcategories of finite-dimensional modules and representations, respectively.

We see $Q^{op}$ appearing in the proposition because of our choice of conventions: right modules, and composition of arrows from right to left. The proof of the proposition is straightforward.

Secondly, over an algebraically closed field, the representation theory of *any* finite-dimensional algebra is governed by a quiver with relations. More precisely:

Assume that the field 