Let $R$ be a noetherian domain with field of fractions $F$, let $V$ be a finite-dimensional $F$-vector space, and let $M,N \subseteq V$ be $R$-lattices in $V$ (finitely generated $R$-submodules of $V$ containing a basis for $V$ over $F$).
We define the $R$-index of $N$ in $M$, written $[M:N]_R$, to be the $R$-submodule of $F$ generated by the set $$\{ \det(\delta) : \delta \in \mathrm{End}_F(V)\text{ and } \delta(M) \subseteq N\}.$$
If $R=\mathbb{Z}$ and $N \subseteq M$, then $[M:N]_{\mathbb{Z}}=\#(M/N)$ is the usual index of abelian groups.
I'm looking for a reference that treats $R$-indices in this level of generality, establishing its basic properties so I don't have to do this myself by hand. Any help would be most appreciated.
