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 CR 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 CR. 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).