The original context of the following question is something about coherent sheaves over noetherian schemes, but the question itself is purely (commutatively-)algebraic.
The definition of an invertible module is usually taken to be "locally free of rank one", namely if $M$ is an $A$-module for some ring $A$, then $M$ is invertible if $M_\mathfrak{p}\cong A_\mathfrak{p}$ for every $\mathfrak{p}\in \operatorname{Spec}(A)$. This definition is also equivalent to another one (that justifies the name), that there is an $A$-module $N$, such that $M\otimes N\cong A$ (and thus, $M$ represents an invertible element in the monoid of isomorphism classes of $A$-modules with tensor product).
I was able to give a prove for this (I think), yet it is a bit long and technical so I was wondering if there is a good reference for a straightforward and elementary as possible proof of this equivalence somewhere in the literature (I couldn't find something that was focused enough), or if anyone is willing to sketch an outline of such proof for me.