Let R be a noetherian domain with field of fractions F, let V be a finite-dimensional F-vector space, and let M,N⊆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

{detIf 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.

**Answer**

**Attribution***Source : Link , Question Author : John Voight , Answer Author : Community*