Introduction

Since the structure theory for modules over arbitrary orders (which are in general not Dedekind domains) is very unsatisfactory, modules over orders in Magma are always modules over some maximal order of a number field or function field.

Let k be a number field or function field and ( O)_k its ring of integers. Since ( O)_k is a Dedekind domain, every finitely generated torsion free module M over ( O)_k has a representation as a direct sum M = sum_(i=1)^m (frac A_i) alpha_i = { sum_(i=1)^m a_ialpha_i | a_i in frac A_i} with (fractional) ideals (frac A_i) and elements alpha_i in kM isomorphic to k^r.

A (not necessarily direct) sum sum _(i=1)^m (frac A_i)alpha_i will be represented as a pseudo--matrix ((frac A)|A) where (frac A) = ((frac A_1), ..., (frac A_m))^t is a column vector of ideals and A = (alpha_1, ..., alpha_m)^t in k^(m x r) is a matrix. The ideals (frac A_i) are called coefficient ideals.

This pseudo--matrix is called a pseudo--basis iff the sum is direct. A pseudo--matrix ((frac A)| A) is in Hermite normal form iff there are s <= m, 1 <= i_1 < i_2< ... < i_s such that A_(j, l) = 0 (1 <= l<i_j), A_(j, i_j) = 1 and A_(j, l) is reduced modulo (frac A_j)(frac A_l)^(-1). For j>s we have A_(j, l) = 0.

This normal form is unique if a suitable reduction is used.

As a consequence of this normalisation, usually alpha_i not in M. To be precise: alpha_i in M iff 1 in (frac A_i).

All modules are in Hermite normal form, i.e. every module is represented by a pseudo--basis in Hermite normal form.

General (non torsion free) modules are represented as quotients of a torsion free module M and a submodule S. Elements of Q := M/S are represented as elements of M, arithmetic in Q is reduced to arithmetic in M followed by a reduction modulo the pseudo--basis of S.