Cohomological left and right differential graded modules

Let $A$ be a cohomological DGA over a commutative ring $R$. A cohomological left differential graded $A$-module is a graded unital left $A$-module $M={⨁}_{q\in \mathbb{Z}}{M}^q$ (with module action as in Unital left and right modules over a ring; unqualified module means left module) whose action has degree zero, $A^p{M}^q\subseteq M^{p+q}$, together with $R$-linear maps $d_M^q:M^q\to M^{q+1}$ satisfying $d_M^{q+1}{d}_M^q=0$ and, for homogeneous $a\in A^p$ and $m\in M^q$, $d_M(am)=d_A(a)m+(-1{)}^{p}a{d}_M(m)\mathrm{.}$

A cohomological right differential graded $A$-module is a graded unital right $A$-module $M$ with $M^q{A}^p\subseteq M^{q+p}$ and a degree-one square-zero $R$-linear differential satisfying, for homogeneous $m\in M^q$ and $a\in A^p$, $d_M(ma)=d_M(m)a+(-1{)}^{q}m{d}_A(a)\mathrm{.}$

The left and right conventions are separate. A differential graded bimodule uses both actions with the commuting-action condition of $(S,R)$-bimodules and commuting left and right scalar actions; both differential compatibility identities then apply. A left module is not silently identified with a right module when $A$ is noncommutative.