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An algebra $A$ over a field (ring) $K$ which is a differential ring and such that, moreover, any derivation $\partial$ commutes with multiplications by elements of $K$, i.e. $\partial(\alpha x) = \alpha \partial(x)$, where $\alpha \in K$, $x \in A$.


Cf. also Derivation in a ring.

A differential graded algebra (or DGA) over a ring $K$ is a graded algebra $A$ equipped with a graded $K$-module homomorphism $\partial : A \rightarrow A$ of degree $-1$ such that $\partial^2 = 0$ and such that $\partial$ is a derivation in the graded sense, i.e. such that $\partial(uv) = \partial(u) v + (-1)^{\deg u} u \partial(v)$. They are of importance in (co)homology theory.


[a1] S. MacLane, "Homology" , Springer (1963)
How to Cite This Entry:
Differential-algebra(2). Encyclopedia of Mathematics. URL:
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article