There is a natural symmetric pairing on *n*×*n* matrices given by

Now, *R* is a subgroup of the finitely-generated free abelian group of
*n*×*n* matrices with integer coefficients; thus *R* is a
finitely-generated free abelian group as well. If is any
element of *K* we can clear denominators to find an integer *d* so
that *d* is a matrix with integer entries. It follows that *R*
contains a basis of *K* as a vector space over
. Thus *R* is of
the form
^{ . }*w*_{1} + ... + ^{ . }*w*_{n}; moreover,
*K* = ^{ . }*w*_{1} + ... + ^{ . }*w*_{n}. Let denote the
collection of all elements in *K* so that
< , > is
an *integer* for all in *R*. Finding such an is
clearly equivalent to solving the system of equations

r_{1}^{ . }w_{1}, w_{1} |
+ | ^{ ... } |
+ | r_{n}^{ . }w_{n}, w_{1} |
= | p_{1} |

r_{1}^{ . }w_{1}, w_{n} |
+ | ^{ ... } |
+ | r_{n}^{ . }w_{n}, w_{n} |
= | p_{n} |

Now suppose
*R* *S*, where *S* is another order (i. e. an *R*_{g}
for some *g*). We clearly have the sequence of inclusions
*R* *S* . It follows that *D*_{S} * divides* *D*_{R}; by decreasing induction we see that there is a
maximal order. We also note that by duality, *S*/*R* and
/ have the same order, so that *D*_{R} is the
multiple of *D*_{S} by the *square* of an integer. Let
_{K} be
the collection of all elements of *K* whose characteristic polynomials
have integer coefficients; one can show that this is closed under
addition and multiplication. It is clear that
_{K} contains *R*
since very matrix with integer entries has a characteristic polynomial
with integer coefficients. By the above, we see that
_{K} is
contained in , hence it is finitely generated; let
_{k} = ^{ . }*u*_{1} + ... + ^{ . }*u*_{n}. Let *v* be any non-zero
column vector and consider the basis
*u*_{i}^{ . }*v* of the space of
column vectors. With this change of basis, each each element of
_{K} is represented by a matrix with integer entries. Thus
_{K}
is an order and the *unique* maximal order.

An extension of the example we looked at for fields is to associate an
order with an irreducible polynomial
*P*(*T*) = *T*^{n} + *a*_{1}*T*^{n - 1} + ... + *a*_{n}
where the *a*_{i} are all integers. We continue the notation of the
previous subsection. It follows that is a matrix with
integer coefficients; with a little effort one can also show that the
natural order *R*_{P} in
() is precisely the collection of
all integer linear combinations of the powers 1, , ...,
. The discriminant of this order is also the
discriminant of the polynomial *P*(*T*) and is denoted as *D*_{P}. Unlike
the case of fields, however, it is *not* true that every order has
the form *R*_{P} for some polynomial *P*(*T*).