The Arithmetic of Quaternion Algebra

1.4. ORDERS AND IDEALS 17
Under the notations of the precedent definitions, we have P I = IP ′ , the
orders O ′ , O are tied, and the prime two-sided ideals P, P ′ belong to the same
model. We denote the model of P by (P ), and define the product (P )I by setting
(P ) = P I = IP ′ . We see at once the product is commutative: (P )I = I(P ).
Proposition 1.4.6. The product of a two-sided ideal J by an ideal I equals the
product JI = IJ ′ , where J ′ is a two-sided ideal belonging to the model of J.
. For example, the maximal orders are tied each other, and the normal ideal
commute with the models of the normal two-sided ideals.
Properties of the non two-sided ideal.
Let O be an order. An integral ideal P to the left order O is said to be irreducible
if it is nonzero, different from O, and maximal for the inclusion in the set of the
integral ideals to the left of the order O.
We leave the verification of the following properties (they had been proved in
Deuring [1], or Reiner [1])as an exercise :
1) P is a maximal ideal in the set of the integral ideal to the right of Or(P ).
2) If O is a maximal order, P contains a unique two-sided ideal of O.
3) If M = O/P , the ideal I = x ∈ O, xM = 0 , the annihilator of M in O, is a
two-sided ideal contained in P ( assuming O to be maximal).
4) an integral ideal is a product of irreducible ideals.
Definition 1.12. The reduced norm n(I) of an ideal I is the fractional ideal of
R generated by the reduced norms of its elements.
If I = Oh is a principal ideal, n(I) = Rn(h). If J = O ′ h ′ is a principal
ideal, to the left of the order O ′ = h −1 Oh, then we have IJ = Ohh ′ and
n(IJ) = n(I)n(J). the last relation remains valid for the non-principal ideals.
For proof it can be utilized that an ideal is finitely generated over R.
One can find the proof in Reiner’s or just to do as an exercise.For the ideals
we shall consider in the following chapters (principal or locally principal), the
the multiplicative of the norm of ideal can be derived from the multiplicative of
the norm over the principal ideals.
Different and discriminant
Definition 1.13. The different O ⋆−1 of an order O is the inverse of the dual of
O by the bilinear form induced by the reduced trace: O ⋆ = {x ∈ H, t(xO) ⊂ R}.
Its reduced norm n(O ⋆−1 ) is called the reduced discriminant of O, denoted by
D(O).
We have the following lemma.
Lemma 1.4.7. (1) Let I be an ideal. The set I ⋆ = {x ∈ H, |t(xy) ∈ R, ∀y ∈ I}
is a two-sided ideal.
(2) Let O be an order. The ideal O ⋆−1 is an integral two-sided ideal.
(3) If O is a free R-module with basis (ui) and a principal ring, then n(O ⋆−1 ) 2 =
R(det(t(uiuj)).
Proof. (1) It is clear, I ⋆ is a R-module . By the analogy with what we used
in the proof of the equivalence of that two definition of orders ( prop. 4.1), we
can prove there exists d ∈ R such that dO ⊂ I ⋆ ⊂ d −1 O , thus I ⋆ is an ideal.
Its left order {x ∈ H|t(xI star I) ⊂ R} equals its right order {x ∈ H|t(I ⋆ xI)},