Some Basic Machine Algorithms for Integral Order Computations

摘要

Three machine implemented algorithms for computing with integral orders are described. The algorithms are: For an integral order R given in terms of its left regular representation relative to any basis, compute the nil radical J(R) and a left regular representation of R/J(R); For a semisimple order R given in terms of its left regular representation relative to any basis, compute a new basis for R and the associated left regular representation of R such that the first basis element of the transformed basis is an integral multiple of the identity element in Q(x)R; Relative to any fixed Z-basis for R, compute a unique canonical form for any given finitely generated Z-submodule of Q(x)R described in terms of that basis. (Author)