# 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 $M_n(R)$.