Module
Given a ring with identity \(R\), a (left) \(R\)-module is an abelian group \((M, +)\) and a mapping \(. : R \times M \to M\) such that for all \(s, r \in R\) and \(m, n \in M\)
- \(1.m = m\)
- \(s.(r.m) = (sr).m\)
- \(s.(m + n) = s.m + s.n\)
- \((s + r).m = s.m + r.m\)
We call this operation scalar multiplication.
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.