Definition 15.102.1. Let $R$ be a ring. An $R$-module $M$ is invertible if the functor

$\text{Mod}_ R \longrightarrow \text{Mod}_ R,\quad N \longmapsto M \otimes _ R N$

is an equivalence of categories. An invertible $R$-module is said to be trivial if it is isomorphic to $A$ as an $A$-module.

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