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).