By collecting the denominators of the generators of *M* we can find a
non-zero integer *d* so that *d*^{ . }*M* is contained in *R*. Since
this is a subgroup of *R* that is closed under multiplication by *R*,
it is an *ideal* *I* in *R*. Thus *M* = *d*^{-1}*I* is a *fractional
ideal* for *R*. It is clear that
*R*(*d*^{ . }*M*) = *R*(*M*) = *R*. More
generally, for any non-zero in *K*, we have
*R*(^{ . }*M*) = *R*. Moreover,
^{ . }*M* is obtained by replacing the *v* in
the previous paragraph by
*v*, which is just another
non-zero vector.

Conversely, let *I* be a non-zero ideal in the ring *R*. Let
be a non-zero element of *I*. Then
is in *K* and by
collecting the denominators we find a non-zero integer *d* so that
*d*^{ . } has integer coefficients so is in *R*. But then
*d* = *d*^{ . } is in *I*; thus *I* contains *d*^{ . }*R*.
In particular, *I* contains a basis of *K* and is a free group of rank
*n*; in other words *I* is a lattice. Clearly *R* is contained in
*R*(*I*) but in general the latter could be bigger.

Now, for any non-zero ideal in *R* we have the *restriction*
*I* = *a*. By the above discussion this is a non-zero ideal in
. We also see that *R*/*I* is a quotient of the finite group
*R*/*aR*; the latter group has order *a*^{n}. The order of *R*/*I* is
called the *norm* of the ideal and denoted as
Nm(*I*). The norm
of an element is
det(); these two definitions are
related since
Nm(^{ . }*R*) = | det()| (Exercise).

Now, we noted above that
Nm(*d*^{ . }*R*) = *d*^{n} for any positive integer
*d* so we can extend the above definition by defining for *M* = *d*^{-1}*I*,
Nm(*M*) = *d*^{-n}Nm(*I*). Similarly, the restriction of *d*^{ . }*R* is
clearly *d*, so we define the restriction of *M* to be
*d*^{-1}(*I* ). When *M* is contained in (i. e. *M* is an ideal)
*R*, the two definitions are consistent.