Two remarks about hereditary orders

In the first remark it is shown that, over a Dedekind ring, hereditary orders in a separable algebra are precisely the “maximal” orders under a relation stronger than inclusion (Theorem 1). At the same time simple proofs for known structure theorems of hereditary orders are obtained. In the second remark a complete classification is given of lattices over a hereditary order, provided the underlying Dedekind ring is contained in an algebraic number field and the lattices satisfy the Eichler condition (Theorem 2).