The Stacks project

Definition 22.4.1. Let $R$ be a ring. Let $(A, \text{d})$ be a differential graded algebra over $R$. A (right) differential graded module $M$ over $A$ is a right $A$-module $M$ which has a grading $M = \bigoplus M^ n$ and a differential $\text{d}$ such that $M^ n A^ m \subset M^{n + m}$, such that $\text{d}(M^ n) \subset M^{n + 1}$, and such that

for $a \in A$ and $m \in M^ n$. A homomorphism of differential graded modules $f : M \to N$ is an $A$-module map compatible with gradings and differentials. The category of (right) differential graded $A$-modules is denoted $\text{Mod}_{(A, \text{d})}$.