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:
- (a)
- Abstract Modules: Given a ring R, a set M and a mapping
φ : R x M -> M, the pair (M, φ) 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.
- (b)
- 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.
- (c)
- Modules with Matrix Action: Let R be a PIR and suppose
S is a R-algebra. Thus there exists a ring homomorphism
φ : R -> S, and so S is a (left) R-module with the R-action
defined by r * s = φ(r) * s. Indeed, any S-module M is a (left) S-module
with action defined by r * m = φ(r) * m. Furthermore, if φ(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 Mn(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 Mn(R).
V2.28, 13 July 2023