Throughout this chapter, when discussing the set of all R-homomorphisms from the R-module M into the R-module N, it will be assumed that R is a commutative ring. We further assume that M and N are free R-modules and that bases for these modules are present. The module Hom_(R)(M, N) will be identified with the module of m x n matrices over R. Thus, an element of Hom_(R)(M, N) is represented as a matrix relative to the bases of the generic modules corresponding to M and N. For this reason, we will refer to these modules as matrix modules.

We remind the reader that submodules of Hom_(R)(M, N) are always presented in embedded form. If the user wishes to have submodules presented in reduced form then he/she should use the natural isomorphism between R^((m x n)) and R^((mn)).

It should be noted that essentially every tuple module operation for elements and submodules applies to matrix modules. Thus, all of the operations of the previous chapter are assumed to apply to matrix modules.

The modules M and N may themselves be matrix modules. In this case, the resulting matrix module has either a right or left action and an element of it transforms a (homomorphism) element of M into a (homomorphism) element of N.

The function Reduce may be used to construct for a matrix module H the matrix module H' equivalent to H whose elements are with respect to the actual bases of the domain and codomain of elements of H (not the generic bases of the domain and codomain).