Differential graded algebra

Remarks

This is the cohomological convention; in the homological convention $d$ lowers the degree by one. The cohomology $H(A) = \ker d / \operatorname{im} d$ is a graded algebra, because clause 3 makes $\ker d$ a subalgebra and $\operatorname{im} d$ a two-sided ideal of it. Example: let $K$ be a field, $A^0 = K[x]$, $A^1 = K[x]\, dx$, and $A^n = 0$ otherwise, with $(dx)^2 = 0$. Put $d(f) = f'\, dx$ on $A^0$ and $d = 0$ on $A^1$. Clauses 1 and 2 are clear, and clause 3 reduces to the product rule $(fg)' = f'g + fg'$; for $a = f$, $b = g\,dx$ both sides are $0$ since $(dx)^2 = 0$. Non-example: in the same algebra put $d(f) = f\, dx$ instead. Clauses 1 and 2 hold, but clause 3 fails: $d(1 \cdot 1) = dx$ while $d(1)\cdot 1 + 1 \cdot d(1) = 2\, dx$. Every graded algebra with $d = 0$ is a differential graded algebra.