General Modules

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 \(\phi : R \times M \rightarrow M\), 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.

(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 \(\phi : R \rightarrow S\), 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 \(\phi(R)\) 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 \(M_n(R)\).