The maximal ideals in quaternion orders

Abstract:

Let $R$ be a Noetherian, integrally closed local domain, and $\Lambda$ an $R$-order in a generalized quaternion algebra over the quotient field of $R$. In this note, it is proved that: (a) $\Lambda$ has at most two maximal ideals; and (b) in case $\Lambda$ does have exactly two maximal ideals, the corresponding residue class rings are commutative fields.