Let [*M* : *R*] denote the collection of all in *K* so that
^{ . }*M* is contained in *R*. Clearly, *d*^{ . }*R* is contained
in [*M* : *R*]. On the other hand
= [*M* : ] was shown
above to contain all that send *M* into . The
latter contains *R* so we see that [*M* : *R*] is contained in
. Thus [*M* : *R*] is also a lattice. Specifically, we define
*C*_{R} as
[ : *R*].

Armed with this result, we now consider the collection of all lattices
*M* with the property that *R*(*M*) = *R* for a fixed *Gorenstein* order
*R*. This collection of lattices includes *R*, and *C*_{R}.
For any such *M*, the above lemma says that
*M*^{ . }[*M* : *R*] = *R*. If we
define the product of *M* and *N* as *M*^{ . }*N*, then this shows that
we have a group with *R* playing the role of identity. It is further
clear that *M* and
^{ . }*M* are naturally isomorphic for any
non-zero in *K*. We may further consider lattices modulo such
isomorphisms. This gives us the *class group* of invertible
fractional ideals modulo isomorphism which is denoted by
Cl(*R*). We
noted above that there could be ideals (and fractional ideals) *M* for
*R* such that *R* is a proper subring of *R*(*M*). In this case we do
not necessarily have *M*[*M* : *R*] = *R*; we do not include such *M* in the
class group. However, since *R*(*M*) is an order as well, this situation
cannot arise if *R* is the maximal order
_{K}. The corresponding
class group is sometimes loosely referred to as the class group of *K*
and denoted
Cl(*K*).