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Given a ring $R$ (Chapter 8), a (left) $R$-module is an abelian group $(M, +, 0, -(-))$ together with a scalar action $R \times M \to M$, written $r \cdot m$, satisfying:

\[\begin{aligned} \text{(M1)}&\quad r \cdot (m + n) = r\cdot m + r \cdot n \\ \text{(M2)}&\quad (r + s) \cdot m = r \cdot m + s \cdot m \\ \text{(M3)}&\quad (r \cdot s) \cdot m = r \cdot (s \cdot m) \\ \text{(M4)}&\quad 1 \cdot m = m \end{aligned}\]

for all $r, s \in R$, $m, n \in M$. This is exactly the vector space definition, with “field” replaced by “ring”. That extra generality is the whole point: $\mathbb{Z}$-modules are abelian groups, and $k[x]$-modules (for $k$ a field) are vector spaces equipped with a chosen linear endomorphism. Most importantly for this book, a representation of a quiver $Q$ is precisely a module over the path algebra $kQ$; this is why the present chapter is placed immediately before Chapter 11.

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