Fractional ideal of a quaternion order

Let $B$ be a quaternion algebra over a number field $K$ , let $R=\mathcal O_K$ , and let $\mathcal O\subseteq B$ be an $R$ -order. A right fractional $\mathcal O$ -ideal is a full $R$ -lattice $I\subseteq B$ with $I\mathcal O\subseteq I$ . A left fractional ideal instead satisfies $\mathcal OI\subseteq I$ .

The side is part of the definition. An ideal need not be contained in $\mathcal O$ ; one contained in $\mathcal O$ is called integral relative to that order.

Principal ideals

For $a\in B^\times$ , the lattice $a\mathcal O$ is a principal right fractional ideal, and $\mathcal Oa$ is a principal left fractional ideal. They need not be the same subset.

Every fractional ideal has a denominator: some nonzero $d\in R$ satisfies $dI\subseteq\mathcal O$ . This follows by expressing finitely many lattice generators in a fraction-field basis of the order.

The fullness condition

In a split matrix algebra, a nonzero one-sided ideal in the ring-theoretic sense need not span the full algebra. For example, $E_{11}M_2(R)$ has only its first row possibly nonzero and is not a full fractional ideal of $M_2(R)$ .

For ideal-class arithmetic one uses locally principal fractional ideals. This is different from the commutative definition of a fractional ideal inside a field.

References

John Voight, Quaternion Algebras, author's text, Definition 16.2.9 and Remark 16.2.10.