Module

Comparing this to the definition of a vector space, it is clear that it is the same, except the set of scalars is a ring rather than a field. In this sense, modules generalise vector fields.

Comparing this to the definition of a $G$-module, we can notice that the first two properties come about by $M$ being an $R^{\ast }$-set, while the third is required for a $G$-module. Thus our only additional condition is that $(r+s).m=r.m+s.m$. In this sense a module is just a group action of a ring on an abelian group where the operation respects the addition of both the ring and the abelian group.