Much of the machinery of modules and vector spaces are fairly straightforward, or discussed above.
An external ring (S, \oplus, \otimes) over the ring (R, +, \cdot) is an abelian group (S, \oplus) and an external monoid (S, \otimes) where \otimes is externally distributive in (S, \oplus, \otimes) over the ring (R, +, \cdot).