Definition 22.28.1. Bimodules. Let $R$ be a ring.

1. Let $A$ and $B$ be $R$-algebras. An $(A, B)$-bimodule is an $R$-module $M$ equippend with $R$-bilinear maps

$A \times M \to M, (a, x) \mapsto ax \quad \text{and}\quad M \times B \to M, (x, b) \mapsto xb$

such that the following hold

1. $a'(ax) = (a'a)x$ and $(xb)b' = x(bb')$,

2. $a(xb) = (ax)b$, and

3. $1 x = x = x 1$.

2. Let $A$ and $B$ be $\mathbf{Z}$-graded $R$-algebras. A graded $(A, B)$-bimodule is an $(A, B)$-bimodule $M$ which has a grading $M = \bigoplus M^ n$ such that $A^ n M^ m \subset M^{n + m}$ and $M^ n B^ m \subset M^{n + m}$.

3. Let $A$ and $B$ be differential graded $R$-algebras. A differential graded $(A, B)$-bimodule is a graded $(A, B)$-bimodule which comes equipped with a differential $\text{d} : M \to M$ homogeneous of degree $1$ such that $\text{d}(ax) = \text{d}(a)x + (-1)^{\deg (a)}a\text{d}(x)$ and $\text{d}(xb) = \text{d}(x)b + (-1)^{\deg (x)}x\text{d}(b)$ for homogeneous elements $a \in A$, $x \in M$, $b \in B$.

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