Subsections

• Taxonomy of Modules
• Presentation of Submodules
• Module Categories
• Notations

A module M is always regarded as a submodule or quotient module of the free module S^((n)), for some ring or algebra S. The types of module that are definable in the system fall into three classes:

• Abstract Modules: Given a ring R, a set M and a mapping phi: R x M -> M, the pair (M, phi) will be referred to as an abstract R-module. Because of the very general nature of this construction, only the basic arithmetic operations may be applied to modules of this type.

• Modules with Scalar Action: Given a general ring R, an R-module with scalar action is a submodule or quotient module of the free R-module R^((n)), where the action is that of ring multiplication in R.

• Modules with Matrix Action: Let R be a PIR and suppose S is a R-algebra. Thus, there exists a ring homomorphism rho : R -> S, and so S is a (left) R-module with the R-action defined by r * s = phi(r) * s. Indeed, any S-module M is a (left) S-module with action defined by r * m = phi(r) * m. Furthermore, if rho(R) lies in the centre of S, then S acts on M as a ring of R-module endomorphisms. Consequently, M is an S-module. We take M to be the free R-module R^((n)) and so the action of S on M is given by the action of a subring of M_n(R) on M. Thus, given an R-algebra S, an S-module of the form M = R^((n)) may be specified by giving M together with a homomorphism of S into M_n(R).

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 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.

Module Categories

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;

• ModTupRng - the category of modules of n-tuples over a ring;

• ModMatRng - the category of modules of m x n matrices over a ring;

• ModGrp - the category of R[G]-modules, where G is a group.

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 (including vector spaces) while U and V will denote vector spaces.