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This chapter describes the facilities provided for modules in the
following representations:
- Tuple Modules: Modules whose elements are n-tuples
over a fixed ring R, i.e., modules R(n);
- Matrix Modules: Modules whose elements are
homomorphisms of modules, i.e., HomR(M, N). The elements of
these modules are m x n matrices over the ring R;
The ring R acts on the right of the module element by scalar
multiplication. If R is not an Euclidean Domain then, currently,
only arithmetic with vectors is supported. In particular, the ability
to work with submodules and quotient modules is restricted to situations
where R is either a field or Euclidean Domain.
In the first part of the chapter we describe the operations that apply
to modules generally, while in the second half we describe the creation of
modules HomR(M, N) together with the operations that are specific to
them. Insofar as elementary module-theoretic operations are concerned,
there is no real difference between tuple modules and matrix modules
except for the input and display of elements. Many special operations
provided for matrices are described in the chapter on matrices.
The reader is referred to the chapter on vector spaces for descriptions
of the extensive functionality provided for modules over fields.
The family of all finitely generated modules over a given ring R
forms a category, while the set of all finitely generated modules forms
a family of categories indexed by the ring R. In this family of
categories, objects are modules and the morphisms are module homomorphisms.
The category name for modules is ModRng.
We distinguish the following subcategories of ModRng:
- ModTupFld - the category of modules of n-tuples over a field;
- ModMatFld - the category of modules of m x n matrices
over a field;
- ModTupEd - the category of modules of n-tuples over an
euclidean domain.
- ModTupRng - the category of modules of n-tuples over a ring;
- ModMatRng - the category of modules of m x n matrices over a ring;
Let N be a free submodule of the R-module M. We have two alternative ways
of presenting N. Firstly, we can present it on a set of generators that
are elements of M; we call such a presentation an
embedded presentation. Alternatively, given that N has rank r, we
can present it as the module S(r), with appropriate action induced
from the action of R on M. We call this presentation of N a reduced
presentation.
The user can control the method of submodule presentation at the time of
creation of an initial module through selection of the appropriate creation
function. Thus, the function RModule will create a module with the
convention that it and all its submodules and quotient modules will have
their submodules presented in reduced form. The use of RSpace, on the
other hand, signifies that submodules are to be presented in embedded form.
Throughout this chapter, R will denote a ring (possibly a field) while K
will denote a field. The letters M and N will denote modules,
while U and V will denote vector spaces.
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