ModulE

A left R-module consists of some commutative group (M,+) together with a RinG of scalars (R,+,*), together with an operation RxM->M (scalar multiplication, usually just denoted *) such that

   For r,s in R, x in M, (rs)x = r(sx)
   For r,s in R, x in M, (r+s)x = rx+sx
   For r in R, x,y in M, r(x+y) = rx+ry

A right R-module is defined similarly, only the ring acts on the right.  The two are easily interchangeable.

The action of an element r in R is defined to be the map that sends each x to rx (or xr), and is necessarily an EndoMorphism of M.  The set of all EndoMorphism''s'' of M is denoted End(M) and forms a RinG under addition and composition, so the above actually defines a HomoMorphism from R into End(M).

This is called a representation of R over M, and is called faithful if and only if the map is one-to-one.  M can be expressed as an R-module if and only if R has some representation over it.  In particular, every commutative group is a module under the IntegerNumbers, and is either faithful under them or some ModularArithmetic.

A module under a field is called a VectorSpace.