Cohomological differential graded algebra

Let $R$ be a commutative ring with unit. A cohomological differential graded algebra over $R$ (cohomological DGA) consists of a $\mathbb{Z}$-graded $R$-module $A={⨁}_{n\in \mathbb{Z}}{A}^n$, an $R$-bilinear associative multiplication $A^p\times A^q\to A^{p+q}$ with a unit $1\in A^0$ and central $R$-action, and $R$-linear maps $d^n:A^n\to A^{n+1}$ such that $d^{n+1}{d}^n=0$. The degree convention and square-zero differential are those of a cochain complex (Cochain complex in an abelian category); the ungraded multiplication and unit satisfy the $R$-algebra convention in Algebras over a commutative ring, central structure maps, and algebra homomorphisms.

For homogeneous $a\in A^p$ and $b\in A^q$, require the signed Leibniz rule $d^{p+q}(ab)=d^p(a)b+(-1{)}^{p}a{d}^q(b)\mathrm{.}$ No symmetry condition $ab=(-1{)}^{pq}ba$ is imposed on homogeneous elements of degrees $p$ and $q$. In particular, this definition fixes the cohomological sign convention and does not assert any formality, derived-category, or Hochschild-cohomology result.

The unit is automatically closed: applying the Leibniz rule to $1\cdot 1$ gives $d(1)=d(1)+d(1)$, so cancellation in the additive group gives $d(1)=0$.