Let $R$ be a commutative ring. Following Cartan (1954) a differential graded algebra $(A, d)$ over $R$ is a $\mathbb Z$-graded $R$-algebra $A$ with a homogeneous $R$-linear endomorphism $d$ of degree $1$ with $d^2 = 0$ satisfying
\[d(a ⋅ b) = d(a) ⋅ b + (−1)^{∣a∣}a ⋅ d(b)\]
for any homogeneous a, b ∈ A of degree $∣a∣$, resp. $∣b∣$. Similarly, a differential graded module is defined as a $\mathbb Z$-graded $A$-module with an endomorphism $\delta$ of degree $1$ and square $0$ satisfying
\[\delta(a ⋅ m) = d(a) ⋅ m + (−1)^{∣a∣}a ⋅ \delta(m)\]
for all homogeneous a ∈ A and m ∈ M .
Until very recently the ring theory of differential graded algebras and differential graded modules remained largely unexplored. The case of acyclic differential graded algebras was completely classified by Aldrich and Garcia-Rozas in 2002 and the case of R being a field and A being finite dimensional was considered by Orlov in 2020, basically with geometric motivations in mind. In a more systematic study I studied basic ring theoretical questions, such as a notion of dg-Jacobson radicals, a dg- Nakayama lemma, Ore localisation of dg-algebras, and dg-Goldie’s theorem. Most interestingly, several standard properties in general ring theory do not generalise, but some do. We give examples, and further classify dg-division rings and dg-separable dg-field extensions, and also a dg-version of the classical Levitzki-Hopkins theorem on artinian respectively semiprimary algebras.