Modules over Orders (New) [HB 65]

Modules over orders of type ModOrd exist only for maximal orders so that the modules are over a Dedekind domain for which the structure theory is greater.

New Features:

• The Module function can create modules over orders from orders, vectors, ideals and tuples of vectors and ideals.

• Submodules and quotients of modules by submodules can be created using the sub and quo constructors. For quotient modules a generalised SmithForm can be computed.

• The BaseRing, Degree, NumberOfGenerators, Determinanant, Dimension and the vectors generating a module can all be retrieved from a module.

• Equality of modules, whether an element lies in a module and whether one module is a subset of another can all be determined. Intersections of modules can also be taken.

• Modules can be multiplied with ideals and added. Modules can also be formed as the product of a module element and an ideal.

• The Basis of a module can be determined as well as its ElementaryDivisors.

• The Dual of a module wrt. scalar products can be computed.

• The SteinitzClass and SteinitzForm of a module can be computed.

• Homomorphisms between modules can be created as well as the hom-module of homomorphisms between two modules. The image and kernel of homomorphisms can be calculated. A module can be tested for being a submodule of another. The morphism of a module into a submodule or quotient module is returned by Morphism.

• Elements of Modules can be created by coercion of a sequence, vector or module element into a module. These elements can be added, subtracted, multiplied and divided by scalars and multiplied by ideals. Equality of module elements can be determined and elements can be represented as sequences.