Definition 10.12.6. An abelian group $N$ is called an $(A, B)$-bimodule if it is both an $A$-module and a $B$-module and for all $a \in A$ and $b \in B$ the multiplication by $a$ and $b$ commute, so $b(an) = a(bn)$ for all $n \in N$. In this situation we usually write the $B$-action on the right: so for $b \in B$ and $n \in N$ the result of multiplying $n$ by $b$ is denoted $nb$. With this convention the compatibility above is that $(ax)b = a(xb)$ for all $a\in A, b\in B, x\in N$. The shorthand $_ AN_ B$ is used to denote an $(A, B)$-bimodule $N$.

Comment #8367 by Laurent Moret-Bailly on

Unless I missed a convention somewhere, all modules are denoted as left modules, so strictly speaking the notation $xb$ is not defined. I understand all rings are commutative, and the right module notation is convenient here, but maybe a remark would be in order.

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